A large pizza at a Pizza Palace costs $11.50 plus $0.90 per topping. The cost for a Large pizza at Tasty Pizza costs $13.25 $0.55 per topping. Let n represent the number of toppings. Let c represent the total cost for the pizza.

a) Write a system of equations to model this scenario
b) then solve the system (using the SUBSTITUTION method) to find the number of toppings where the cost is the same. Be sure to **show all work**

If the shaded area of a circle is 25 cm squared. if the diameter of the circle is 10cm, the percentage of the circle is shaded is: 31.83% .

We can start by finding the area of the entire circle. The formula for the area of a circle is:

A = πr^2

where A is the area and r is the radius. Since the diameter of the circle is 10 cm, the radius is half of that, or 5 cm. Therefore, the area of the entire circle is:

A = π(5 cm)^2 = 25π cm^2

Next, we can find what percentage of the circle is shaded by dividing the shaded area by the total area of the circle and multiplying by 100:

Percentage shaded = (shaded area / total area) x 100

Percentage shaded = (25 cm^2 / 25π cm^2) x 100

Percentage shaded = 100 / π ≈ 31.83%

Therefore, approximately 31.83% of the circle is shaded.

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the 11th term in the given geometric sequence is equal to [tex]-9216x^{52}[/tex]

A sequence is a list of numbers indexed by the natural numbers 1,2,3,4,5.. A common notation for a sequence is [tex]a_1,a_2,a_3,\cdots[/tex]. Most often for interesting sequences there is some pattern in the list of numbers, and there is some formula to generate the nth number in the sequence. A sequence is called a geometric sequence if the next number number is got from the previous number by multiplying it by a fixed number r. This number is called the common ratio r of the sequence. So if the first number is a, then the second number is ar, the third is [tex]ar^2[/tex]... So that the nth number [tex]a_n = ar^{n-1][/tex].  

For example 1 ,3,9,27,81,.... The sum of the first n terms of such a sequence is equal to [tex]$S_n = \frac{a(r^n-1)}{r-1}$[/tex]. if the absolute value of r, [tex]|r| < 1[/tex], then the sum to infinity is [tex]$S = \frac{a}{1-r}$[/tex].

In our question we are given a geometric sequence whose first term :

[tex]$a = -9x^2\,,\textrm{ and common ratio : r } = 2x^5$[/tex] .

So the 11th term is [tex]$ar^{10} = (-9x^2){(2x^5)}^{10} = (-9x^2)(1024x^{50}) = -9216x^{52}$[/tex]

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Answer: Circumference = 18 π cm

Step-by-step explanation:

We can use this formula to find the circumference of a circle:

➜ r is equal to the radius, also known as half the width of a circle.

      C = 2πr

We will substitute our known values and solve for C by multiplying. Since the answer option includes π in our units, we do not multiply this into the 18.

      C = 2πr

      C = 2π(9)

      C = 18 π cm

Answer: 18π

Step-by-step explanation:

The circumference of a circle is equal to 2πr, with r being equal to the radius. The radius of a circle is defined as the length from any point on the circle to its middlemost point. In this case, we are given the value of the radius as defined by the line. The radius equals 9, which we can plug into the equation.

C= 2πr

C = 2π(9)

C = 18π = 56.55

Hope this helps!

Answer:

Step-by-step explanation:

Rolling a number greater than 7 on a number cube with sides numbered 1 through 6: The probability of rolling a number greater than 7 on a number cube with sides numbered 1 through 6 is 0, as it is not possible to roll a number greater than 6 on a standard six-sided die.

Picking the 8 of Hearts from a standard deck of playing cards: The probability of picking the 8 of Hearts from a standard deck of playing cards is 0, as there is no 8 of Hearts in a standard deck of playing cards.

A spinner with 6 equal sections labeled 1 through 6 lands on a number less than 7: The probability of a spinner with 6 equal sections labeled 1 through 6 landing on a number less than 7 is 1, as all the sections are labeled with numbers less than 7.

Flipping a coin and having it land with the heads side facing up: The probability of flipping a coin and having it land with the heads side facing up is 0.5 or 50%, assuming the coin is fair and has two equally likely sides.

Rolling a sum of 14 with two number cubes with sides 1-6: The probability of rolling a sum of 14 with two number cubes with sides 1-6 is 0, as the highest possible sum that can be obtained by rolling two number cubes with sides 1-6 is 12 (6 + 6).

What is the probability of rolling an even number on a dice? The dice is numbered 1-6: The probability of rolling an even number on a dice numbered 1-6 is 0.5 or 50%, as there are three even numbers (2, 4, and 6) and three odd numbers (1, 3, and 5) on a standard six-sided die.

The following M & M colors are in the bowl: 4 yellow, 6 orange, 3 green, 5 blue, 2 brown. What is the probability of selecting a blue candy? The probability of selecting a blue candy from the bowl is 5/20 or 1/4, as there are 5 blue candies in the bowl out of a total of 20 candies.

A number cube is labeled 1 through 6. The probability of randomly rolling a 4 is 1 over 6. What is the probability of not rolling a 4? The probability of not rolling a 4 on a number cube labeled 1 through 6 is 5/6, as there are 5 possible outcomes that are not 4 (1, 2, 3, 5, and 6) out of a total of 6 possible outcomes.

Picking a purple ball from a bag containing 7 purple balls and 1 green ball: The probability of picking a purple ball from a bag containing 7 purple balls and 1 green ball is 7/8, as there are 7 purple balls out of a total of 8 balls in the bag.

im sorry I don't know how to delete this but i now realize that my awnser was completely wrong.

p=8h is the equation.This part is represented in tabular form.The coefficient of h. Mark worked 21 hours that week.Yes, the equation is a direct variation.

An equation is a statement that shows the equality of two expressions. It typically contains one or more variables and may involve mathematical operations such as addition, subtraction, multiplication, and division. Equations are commonly used in mathematics, science, and engineering to model real-world situations and solve problems.

Direct variation is a mathematical relationship between two variables, where a change in one variable results in a proportional change in the other variable. In other words, if two variables are in direct variation, when one variable increases, the other variable increases as well, and when one variable decreases, the other variable decreases as well, in a constant ratio.

Part A: The equation for this situation is p = 8h, where p represents the pay in dollars and h represents the number of hours worked.

Part B:

| Worked (hrs) | Pay (p) |

|------------------|---------|

| 1                | 8       |

| 2                | 16      |

| 3                | 24      |

| 4                | 32      |

| 5                | 40      |

| 6                | 48      |

and so on.

Part C: The coefficient of h, which is 8, shows the number of hours Mark worked.

Part D: We can use the equation p = 8h and substitute p = 168 to find the number of hours Mark worked.

168 = 8h

h = 21

Therefore, Mark worked 21 hours that week.

Part E: Yes, the equation is a direct variation because the pay (p) is directly proportional to the number of hours worked (h) and the constant of variation is 8.

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The scale factor of the dilation must be greater than 1. if the image A'B'C' is bigger than the preimage ABC

If the image A'B'C' is bigger than the preimage ABC, then the scale factor of the dilation must be greater than 1.

In other words, the length of any segment in the image is larger than the corresponding segment in the preimage by the same factor.

Since D is the center of the dilation, we can use the ratio of the corresponding side lengths of the image and preimage triangles to find the scale factor.

For example, if we want to find the scale factor for A', we can use:

scale factor = B'C'/BC

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A congruence transformation that maps triangle RST to triangle UVW is a reflection along the line y = 1, followed by a translation 2 units right and down by 1 units.

A congruence transformation that maps triangle RST to triangle UVW is a rotation of 180°, followed by a translation 2 units left and down by 5 units.

In Mathematics and Geometry, a transformation is the movement of a point from its initial position to a new location. This ultimately implies that, when a geometric figure or object is transformed, all of its points would also be transformed;

In Mathematics and Geometry, a reflection can be defined as a type of transformation which moves every point of the geometric figure such as a triangle, by producing a flipped, but mirror image of the geometric figure.

By critically observing the geometric figures, we can reasonably infer and logically deduce that the pre-image underwent a sequence of congruence transformation to produce the image.

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The Probability that a student will complete the exam 0.2676.

The probability of completing the exam in one hour or less is:

[tex]P (x < 60)[/tex]

= [tex]P (z < (60-83)/13)[/tex]

=[tex]P (z < -1.77)[/tex]

= 0.0384.

The probability that a student will complete the exam in more than 60 minutes, but less than 75 minutes is.

[tex]P (60 < x < 75)[/tex]

= [tex]P (x < 75)-P (x < 60)[/tex]

Now,

[tex]P (x < 75)[/tex]

=[tex]P (z < (75-83)/13)[/tex]

= [tex]P (z < -0.62)[/tex]

=0.2676.

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More people that rode the roller coaster are between the ages of 11 and 30, than people that are between the ages of 41 and 60 is the best description of the data which is obtained by using the arithmetic operations.

What are arithmetic operations?

Any real number may be explained using the four basic operations, also referred to as "arithmetic operations." Operations like division, multiplication, addition, and subtraction come before operations like quotient, product, sum, and difference in mathematics.

We are given a chart. From the chart we get the following data:

Number of people aged 11 - 20 riding roller coaster = 20

Number of people aged 21 - 30 riding roller coaster = 15

Number of people aged 31 - 40 riding roller coaster = 10

Number of people aged 41 - 50 riding roller coaster = 5

Number of people aged 51 - 60 riding roller coaster = 0

Using addition operation, we get

Total people = 20 + 15 + 10 + 5 + 0

Total people = 40

Now,

Total riders between 11 - 30 = 20 + 15

Total riders between 11 - 30 = 35

Similarly,

Total riders between 41 - 60 = 5 + 0

Total riders between 41 - 60 = 5

Hence, the fourth option is the correct answer.

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The average distance of electric cars is 275.

The range of the data set is 300.

Given:

To find: Average distance of electric cars

Solution:

The average distance of electric cars can be calculated by finding the average of the data set containing the distances of electric cars. Let's assume that the data set is as follows:

250, 250, 250, 400, 300, 300, 350, 100

To find the average distance of electric cars, we can use the formula:

Average = (Sum of all the data points) / (Number of data points)

Substituting the given values, we get:

Average = (250 + 250 + 250 + 400 + 300 + 300 + 350 + 100) / 8

Average = 2200 / 8

Average = 275

Therefore, the average distance of electric cars is 275.

Now, let's calculate the range of the data set. The range is the difference between the maximum and minimum values in the data set. From the given data set, we can see that the minimum value is 100 and the maximum value is 400.

Range = Maximum value - Minimum value

Range = 400 - 100

Range = 300

Therefore, the range of the data set is 300.

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The theorem is shown in a pythagoras theorem is c² = a² + b²

The theorem shown in the Pythagorean theorem is "a² + b² = c²". This theorem is named after the ancient Greek mathematician Pythagoras, who is credited with discovering it.

The Pythagorean theorem applies to right-angled triangles and states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Mathematically, we can express this as:

c² = a² + b²

Where c is the length of the hypotenuse, and a and b are the lengths of the other two sides (called the legs) of the right-angled triangle.

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The results of the balls in Phillip's bag are:

The mean = 8.

The median = 7.

The mode = blue

The range = 12

The mean (or the average) is the sum of all the values divided by the total number of values. Let's calculate the mean for the given data:

Total number of balls = 8 + 3 + 6 + 3 + 13 + 15 = 48

Mean = (8 + 3 + 6 + 3 + 13 + 15) / 6 = 48 / 6 = 8

Mean = 8.

The median is the middle value when a set of values is arranged in ascending or descending order.

Let's arrange the given data in ascending:

3, 3, 6, 8, 13, 15

As the total number of values is even, the median will be the average of the two middle values, which are 6 and 8.

Median = (6 + 8) / 2 = 7

The median = 7.

The mode is the value that appears most frequently in a set of values. Let's find the mode of the given data:

Red balls: 8

Green balls: 3

Yellow balls: 6

Orange balls: 3

Black balls: 13

Blue balls: 15

Blue balls have the highest frequency (i.e., 15) among all the colors.

The range is the difference between the highest and lowest values in a set of values. Let's find the range of the given data:

Highest value = 15 (blue balls)

Lowest value = 3 (green balls and orange balls)

Range = Highest value - Lowest value = 15 - 3 = 12

Range = 12.

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Answer:

Harry can buy 5 gallons of lemonade and 5 gallons of iced tea and still have $1.50 left over from his original $24 dollar budget

Step-by-step explanation:

We can see that they are similar because the ratio of the corresponding sides of the triangle are the same = 5/3.

A triangle is a polygon with three sides and three angles. It is one of the most basic shapes in geometry and is formed when three non-collinear points are connected by straight lines. The three sides of a triangle can have different lengths, and the three angles can have different measures. Triangles can be classified based on their sides and angles.

BC/DE = AD/AB

15/9 = AB/6

AB = (15 × 6)/9 = 90/9 = 10.

15/9 = 10/6 = 5/3.

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The solution to the IVP given by y''+y=t, y(0)=1, y'(0)=-2 is y(t) = cos(t) - (3/2) sin(t) + (1/2) t.

To solve the Initial Value Problem, we can use the method of undetermined coefficients, which involves assuming a particular form for the solution to the non-homogeneous equation y'' + y = t, and then finding the coefficients of the terms in that form by substituting it back into the equation.

First, we find the general solution to the homogeneous equation y'' + y = 0

The characteristic equation is r² + 1 = 0, which has solutions r = ±i. Therefore, the general solution to the homogeneous equation is

y_h(t) = c₁ cos(t) + c₂ sin(t),

where c₁ and c₂ are constants determined by the initial conditions.

Next, we assume a particular form for the non-homogeneous solution, based on the form of the right-hand side t. Since t is a linear function, we assume that the particular solution has the form

y_p(t) = a t + b.

Substituting this into the differential equation, we get

y''_p + y_p = t

2a + (at+b) = t.

Equating coefficients, we get

a = 1/2, b = 0.

Therefore, the particular solution is

y_p(t) = (1/2) t.

The general solution to the non-homogeneous equation is then the sum of the homogeneous and particular solutions

y(t) = y_h(t) + y_p(t)

= c₁ cos(t) + c₂ sin(t) + (1/2) t.

To determine the constants c₁ and c₂, we use the initial conditions:

y(0) = c₁ cos(0) + c₂ sin(0) + (1/2) (0) = c₁ = 1,

y'(0) = -c₁ sin(0) + c₂ cos(0) + (1/2) (1) = c₂ - (1/2) = -2,

so c₂ = -3/2.

Therefore, the solution to the IVP is

y(t) = cos(t) - (3/2) sin(t) + (1/2) t.

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The equation with the new substitutions and conversion factor is a'B'T' = (1/6) * (2/3)^(1/3) * a^(1/3) * d^(1/2)*a^6/6 where k is (1/6) * (2/3)^(1/3) * a^(1/3) * d^(1/2) The intercept value of the log-log graph would have been -1.108.

Starting with Equation 4: T = ka^6/6, we can substitute a = (d/k)^(1/6) and b = (2/3)^(1/2) * (d/k)^(1/3) to get

T = k[(d/k)^(1/6)]^6/6

T = k(d/k)^(1/2)/6

T = (k^(1/2)/6) * d^(1/2)

Now we can rearrange the equation so that a, b, and t are on the left side

a'B'T' = ka^6/6

(a/d)^(1/6) * b^(2/3) * T' = k[(a/d)^(1/6)]^6/6 * T'

(a/d)^(1/6) * (2/3)^(1/2) * (d/k)^(1/3) * T' = (k^(1/2)/6) * d^(1/2) * T'

(2/3)^(1/2) * (a/d)^(1/6) * d^(1/3) * T' = (k^(1/2)/6) * d^(1/2) * T'

(2/3)^(1/2) * (a/d)^(1/6) * d^(1/3) = k^(1/2)/6

k = [(2/3)^(1/2) * (a/d)^(1/6) * d^(1/3)]^2/6

k = (1/6) * (2/3)^(1/3) * a^(1/3) * d^(1/2)

With this new conversion factor, the intercept of the log-log graph would have changed. The intercept represents the value of T when a = 1 (since log(1) = 0). Using the new conversion factor, we have

T = (1/6) * (2/3)^(1/3) * d^(1/2) * a^(1/3)

T = (1/6) * (2/3)^(1/3) * d^(1/2)

log(T) = log[(1/6) * (2/3)^(1/3) * d^(1/2)]

log(T) = log(1/6) + log[(2/3)^(1/3)] + log(d^(1/2))

log(T) = log(1/6) + (1/3) * log(2/3) + (1/2) * log(d)

So the intercept of the log-log graph would be log(1/6) + (1/3) * log(2/3) = -1.108, assuming that d is held constant. This intercept represents the value of log(T) when log(a) = 0, or when a = 1. In other words, when a = 1, the predicted value of T would be 0.162 (or 16.2% of its maximum value).

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--The given question is incomplete, the complete question is given

" Plug the two substitutions into Equation 4 (T = ka^6/6)). Rearrange the equation so that a, b,t are on the left side of the equation and d remains on the right side, e.g. a'B'T' = ka^6/6 you will figure out what the "k" if you used this version of the equation (including your new conversion factor, how would this have changed the intercept of your log-log graph? what would its value have been?"--

If x is a non-negative real number, then the expression √ x is called the Principal square root. The answer is Principal Square Root.

What is Principal Square Root?

The positive square root of a non-negative real integer is the principal square root. It is the one and only non-negative real number that, when squared, yields the given non-negative real number. The principal square root of 9 is 3, for example, because 3 multiplied by itself equals 9.

The radical sign (√) is frequently used to represent the principal square root. The term √x denotes Principal square root of x. The sign can be used to indicate the negative square root of x in some settings, but this is less frequent.

The principal square root notion is significant in mathematics, including algebra, geometry, and calculus. It's used to solve equations, simplify expressions, and calculate lengths and distances. It's also used in a wide range of scientific and engineering applications.

Therefore, if x is a non-negative real number, then the expression √ x is called the Principal square root.

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If x is a nonnegative real number, then the expression √ x is called the square root of x. The square root of a nonnegative real number is a value that, when multiplied by itself, equals the original number.

If x is a nonnegative real number, then the expression √x is called the "principal square root" of x. The principal square root of a nonnegative real number x is the nonnegative real number that, when multiplied by itself, equals x. In other words, if y = √x, then y * y = x. This ensures that the result is also a nonnegative real number.

For example, the square root of 4 is 2 because 2 multiplied by 2 equals 4. The square root function is denoted by the symbol √ and is used to find the positive number that, when squared, gives the input value.

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∠A = ∠C in trapezoid ABCD with arcAB = arcCD, can be proven with the property of isosceles triangles.

Since arcAB = arcCD, the lengths of the two arcs are equal. This implies that the lengths of the segments subtended by these arcs, AB and CD, are also equal.

Let E and F be the midpoints of the non-parallel sides AD and BC, respectively. Connect E and F with a line segment EF.

Since E and F are midpoints, DE = EA and BF = FC. In addition, since AB = CD = L, we can say that:

DE + EA = BF + FC

EA = FC

So, by the Hypotenuse-Leg (HL) theorem of congruence, triangles AEF and CFE are congruent:

ΔAEF ≅ ΔCFE

Now, since the triangles are congruent, their corresponding angles are equal:

∠A = ∠C

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The sample space are:

{AR, AS, AT, AE, RA, RS, RT, RE, SA, SR, ST, SE, TA, TR, TS, TE, EA, ER, ES, ET}

The favorable outcomes care:

{RS, RT, SR, ST, TR, TS}

The probability is  0.3 or 30%

Part A:

The sample space represents all possible outcomes that can occur when two cards are randomly selected without replacement from the given pile of cards.

The sample space can be represented as follows:

{AR, AS, AT, AE, RA, RS, RT, RE, SA, SR, ST, SE, TA, TR, TS, TE, EA, ER, ES, ET}

Part B:

The favorable outcomes are those outcomes in which both cards are consonants. In this case, the consonants are R, S, and T.

The favorable outcomes can be represented as follows:

{RS, RT, SR, ST, TR, TS}

Part C:

To calculate the probability of selecting 2 cards that are consonants, we need to find the ratio of favorable outcomes to the sample space.

The number of favorable outcomes is 6, and the size of the sample space is 20.

Therefore, the probability of selecting 2 cards that are consonants is:

P(consonants) = favorable outcomes / sample space

P(consonants) = 6 / 20

P(consonants) = 0.3 or 30%

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Step-by-step explanation:

4^-3   =   1/4^3   =   1/64

Thus, the number of current lawnmowers in the Springtown Hardware is 864,900.

In mathematics, a percentage is a number or ratio that may be expressed as a fraction of 100. The Latin word "per centum," which meaning "per 100," is where the word "percent" comes from. % is the symbol used to represent percentages.

When a number is expressed in decimal form, you can calculate its percentage by multiplying it by 100. For instance, multiplying 0.5 by 100 gives you the percentage 50%.

Given data:

Springtown Hardware's inventory - 697,500 lawnmowers.

Current inventory of 24% more lawnmowers.

Current inventory = old inventory  + 24% of old inventory

Current inventory =  697,500 + 24 % of  697,500

Current inventory =  697,500 + 24 *  697,500/ 100

Current inventory =  697,500 + 24* 6,975

Current inventory = 864,900

Thus, the number of current lawnmowers in the Springtown Hardware is 864,900.

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Thus, the appropriate measure of variability found for the given data is 10.

The difference here between maximum and smallest values in a data set is known as the range. Utilizing the exact same units as the data, it measures variability. More variability is shown by larger values.

Given length of ribbons:

3, 6, 9, 11, 12, 13

Minimum value = 3

Maximum value = 13

Actually, the range, which is determined by deducting the lowest value from the greatest value in the dataset, would be the proper measure of variability for this data.

Range = Maximum value - Minimum value

Range = 13 - 3

Range = 10

Thus, the appropriate measure of variability found for the given data is 10.

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Therefore (t) can  be written as **(2 - 3Q) / (Q + 4)**.

Within the given equation, Q is a variable. It stands for an unknowable amount or value that can be ascertained by equation-solving.

A quantity that can take on any one of a number of values is referred to as a variable. A variable in mathematics is a symbol or letter that denotes an unknowable amount or value. The variable Q in the given equation stands for an unknowable quantity or value that can be ascertained by resolving the equation.

The equation is as follows:

Q = (2 - 4t) / (t + 3)

When we divide both sides by (t + 3), we obtain:

Q(t + 3) = 2 - 4t

When we increase the left side of the equation, we obtain:

Qt + 3Q = 2 - 4t

The result of adding 4t to both sides of the equation is:

Qt + 3Q + 4t = 2

Combining related concepts gives us:

t(Q + 4) = 2 - 3Q

The result of dividing both sides by (Q + 4) is:

t = (2 - 3Q) / (Q + 4)

t can therefore be written as **(2 - 3Q) / (Q + 4)**.

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The volume of the prism is gotten as 312 cm³.

Volume is described as a measure of three-dimensional space. It is often quantified numerically using SI derived units or by various imperial or US customary units.

The basic formula for volume is length × width × height.

The formula for the volume of triangular prism = area of triangle x length

area of triangle = 1/2 x base x height

⇒ area of triangle = 1/2 x 8 x 6 = 24 cm²

Therefore, volume of the prism = 24 x 13 = 312 cm³

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Answer:

Surface area of prism A is approximately 12.4 square units.

Step-by-step explanation:

Since prism A and prism B are similar, their corresponding side lengths are proportional. Let's use the scale factor k to represent the ratio of the side lengths of prism B to those of prism A. Then, the volume of prism B is (k³)(27) = 27k³ cubic units. Similarly, the surface area of prism B is (k²)(surface area of prism A).

We are given that the volume of prism B is approximately 512 cubic units. Therefore, we can solve for k:

27k³ = 512

k³ = 512/27

k ≈ 3.62

Now, we can use k to find the surface area of prism A:

surface area of prism B = (k²)(surface area of prism A)

128 = (3.62²)(surface area of prism A)

surface area of prism A ≈ 12.4 square units

Therefore, the surface area of prism A is approximately 12.4 square units.

The number of bacteria after 5 hours is approximately 1013.8.

We can use the formula for exponential growth to solve this problem. If a population grows at a rate proportional to its size, then we can writ

N(t) = N₀ × e^(rt),

where N(t) is the size of the population at time t, N₀ is the initial size of the population, r is the growth rate, and e is the mathematical constant approximately equal to 2.71828.

We know that the culture starts with 40 bacteria, so N₀ = 40. We also know that after 2 hours, the size of the population is 180. We can use this information to solve for r:

180 = 40 × e^(2r)

180/40 = e^(2r)

ln(180/40) = 2r

r = ln(180/40)/2

r ≈ 0.6931

Now we can use the formula to find the size of the population after 5 hours:

N(5) = 40 × e^(0.6931×5)

N(5) ≈ 1013.8

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9. The volume of the triangular pyramid is given by:

V = (1/3)Bh

Here, B is the base area.

The base is shaped as a right triangle thus, using the Pythagoras Theorem we have:

h = sqrt(26.7² - 11.7²) = 23.274 km

The area if the base is:

B = (1/2)bh = (1/2)(11.7 km)(23.4 km) = 136.89 sq. km

Now, the volume is:

V = (1/3)(136.89)(15) = 2053.35 cubic km

Hence, the volume of the triangular pyramid is 2053.35 cubic km.

9. The volume of the triangular pyramid is 2053.35 cubic km. 11. The area of the shaded portion is 348.19 cubic in 13. The slant height of the cone is 8.53 meters.

A fundamental conclusion in geometry relating to the lengths of a right triangle's sides is known as Pythagoras' theorem. According to the theorem, the square of the length of the hypotenuse, the side that faces the right angle, in any right triangle, equals the sum of the squares of the lengths of the other two sides, known as the legs.

9. The volume of the triangular pyramid is given by:

V = (1/3)Bh

Here, B is the base area.

The base is shaped as a right triangle thus, using the Pythagoras Theorem we have:

h = √(26.7² - 11.7²) = 23.274 km

The area if the base is:

B = (1/2)bh = (1/2)(11.7 km)(23.4 km) = 136.89 sq. km

Now, the volume is:

V = (1/3)(136.89)(15) = 2053.35 cubic km

Hence, the volume of the triangular pyramid is 2053.35 cubic km.

11. The volume of a cone is given by:

V = (1/3)πr²h

The dimension of the bigger cone is radius is 9 in, and height 15 in:

V1 = (1/3)π(9 in)²(15 in) = 381.7 cubic in

The dimension of the smaller cone is radius is 4 in, and height 10 in:

V2 = (1/3)π(4 in)²(10 in) = 33.51 cubic in

Now, the area of the shaded portion is:

V1 - V2 = 381.7 - 33.51  = 348.19

13. The volume of a cone is given by:

V = (1/3)πr²h

Substituting the values we have:

542.87 = (1/3)π(6 m)²h

h = 542.87 / [(1/3)π(6 m)²] = 6.05 m

Now, using the Pythagoras Theorem for the slant height we have:

s² = r² + h²

s² = (6 m)² + (6.05 m)²

s² = 72.9

s = √(72.9) = 8.53 m

The slant height of the cone is 8.53 meters.

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The rate at which the base of the triangle is changing when the altitude is 20 cm and the area is 120 cm^2 is 0 cm/s.

To solve this problem, we need to use the formula for the area of a triangle, which is:

A = (1/2)bh

Where A is the area, b is the base, and h is the altitude.

We know that the area is 120 cm^2 and the altitude is 20 cm. So we can plug in these values and solve for the base:

120 = (1/2)b(20)

240 = 20b

b = 12 cm

Now we need to differentiate both sides of the equation with respect to time (t):

dA/dt = (1/2)(db/dt)h

We are given the value of dh/dt (which represents the rate at which the altitude is changing) is 3 cm/s. We need to find db/dt (which represents the rate at which the base is changing).

Plugging in the values we know, we get:

dA/dt = (1/2)(db/dt)(20)

Solving for db/dt, we get:

db/dt = (2dA/dt)/h

Plugging in the values we know, we get:

db/dt = (2)(0)/20

db/dt = 0 cm/s
Since the area is constant (120 cm²) and the altitude is constant (20 cm), the base of the triangle is not changing. Therefore, the rate at which the base is changing is: 0 cm/s

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Answer: To solve for c and v, we need to use the equation:

4c + 6v = 80

We have two variables and one equation, so we need another equation to solve for c and v. We can use the fact that the number of students transported by the drivers is 80. Since each car can seat 4 students and each van can seat 6 students, we can write:

4c + 6v = 80

c + v = number of drivers

Since we want to minimize the number of drivers, we want to minimize the value of c + v. We can use substitution to solve for c and v in terms of one variable. Solving for c in terms of v from the first equation, we get:

4c + 6v = 80

4c = 80 - 6v

c = (80 - 6v)/4

Substituting this expression for c into the second equation, we get:

c + v = number of drivers

(80 - 6v)/4 + v = number of drivers

Multiplying both sides by 4 to clear the fraction, we get:

80 - 6v + 4v = 4(number of drivers)

80 - 2v = 4(number of drivers)

20 - 0.5v = number of drivers

We want to minimize the value of c + v, so we want to minimize the value of v. Since v must be a whole number, we want to find the smallest integer value of v that satisfies the equation. We can plug in integer values of v and find the corresponding value of c, and check if the sum of c and v is less than or equal to the current minimum. Starting with v = 1, we get:

v = 1 -> c = 19.5 -> c + v = 20.5

v = 2 -> c = 18.0 -> c + v = 20.0

v = 3 -> c = 16.5 -> c + v = 19.5

v = 4 -> c = 15.0 -> c + v = 19.0

v = 5 -> c = 13.5 -> c + v = 18.5

v = 6 -> c = 12.0 -> c + v = 18.0

The smallest integer value of v that satisfies the equation is v = 6, which gives c = 12. Therefore, we need 12 cars and 6 vans to transport 80 students.

Step-by-step explanation: