Solve the system
y=2x-2
y=2x+9
​

The number of mangoes she picked is 2.

Number is the concept of a mathematical object used to count, measure, and label. It is an abstract concept, though most familiar in the form of numerals such as "3" or "10". Numbers allow us to compare and manipulate amounts, sequences, shapes, and patterns. They also allow us to create models and make predictions. As a result, they are essential tools in a variety of disciplines, from mathematics and physics to engineering and economics.

Let x be the number of mangoes in each group on Monday. The expression representing the number of mangoes in each group is x + 4/2 = x + 2.

b) Let y be the number of mangoes in each group on Tuesday. The expression representing the number of mangoes in each group is 2y + 2/3 = 2y + 2/3.

c) Let z be the number of mangoes left after subtracting the number of mangoes in each group on Tuesday from the number of mangoes in each group on Monday. The equation representing this information is x + 2 - (2y + 2/3) = z.

d) Substituting the expression for x and y in the equation, we get: (2y + 2/3) + 2 - (2y + 2/3) = z.

Solving the equation, we get z = 2.

Hence, the number of mangoes she picked is 2.

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Step-by-step explanation:

Let's call the original length of the skirt "l". After cutting 7 inches off the bottom, the remaining length of the skirt is l-7. Then, when Christi adds a 5-inch ruffle, the final length of the skirt becomes:

(l-7) + 5

Simplifying this expression, we get:

l - 2

Therefore, the expression that best represents the final length of the skirt is:

O2 - s

The arc angle QPS is 258 degrees.

A cyclic quadrilateral is a four sided shape(quadrilateral) that can be inscribed into a circle. The sum of angles in a cyclic quadrilateral is 360 degrees.

The opposite angles of a cyclic quadrilateral have a total of. 180 ° . In other words, the opposite angles of a cyclic quadrilateral is supplementary angles.

Therefore,

m∠P = 4x - 1

m∠R = 12x - 27

Hence,

4x - 1 + 12x - 27 = 180

16x - 28 = 180

16x = 180 + 28

16x = 208

x = 208 / 16

x = 13

m∠R = 12(13) - 27

m∠R = 156 - 27

m∠R = 129 degrees

Therefore,

m∠R = 1 / 2 (QPS)

129 = 1 / 2 × x

x = 129 ×2

x = 258 degrees

Hence,

mQPS = 258 degrees

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Answer:

hole at (- 1, - 4 )

Step-by-step explanation:

If f(x) has terms that cancel , on the numerator/ denominator then a discontinuity  corresponding to that factor is removable and there is a hole.

f(x) = [tex]\frac{x^2-2x-3}{x^2+3x+2}[/tex] ← factor numerator and denominator

    = [tex]\frac{(x-3)(x+1)}{(x+2)(x+1)}[/tex] ← cancel (x + 1) on numerator/ denominator

   = [tex]\frac{x-3}{x+2}[/tex]

this indicates that x + 1 = 0 ( or x = - 1) is a removable discontinuity and the graph has a hole in it.

substitute x = - 1 into the remaining f(x) for y- coordinate of hole

f(- 1) = [tex]\frac{-1-3}{-1+2}[/tex] = [tex]\frac{-4}{1}[/tex] = - 4

coordinates of hole are (- 1, - 4 )

Answer:

Yes, this function has one hole

This occurs at [tex](-1, -4)[/tex]

Step-by-step explanation:

Definition of hole
A hole on a rational function represents the fact that the function approaches the point, but is not actually defined on that precise x value.

Given function
[tex]f(x) = \dfrac{x^2-2x\:-3}{x^2+\:3x\:+\:2}\\[/tex]

Factor the numerator:
[tex]x^2 - 2x - 3 = (x+1)(x - 3)[/tex]

Factor the denominator:
[tex]x^2+\:3x\:+\:2 = (x + 1)(x + 2)[/tex]

Hence

[tex]\begin{aligned}f(x) & = \dfrac{x^2-2x\:-3}{x^2+\:3x\:+\:2}\\\\& = \dfrac{ (x+1)(x - 3)}{(x + 1)(x + 2)}\\\\ &= \dfrac{x - 3}{x+2} \end{aligned}[/tex]

The common factor is   [tex](x + 1)[/tex]

Set this = 0 to find the hole:
[tex]x + 1 = 0 = > x = -1[/tex]

So at[tex]x = -1[/tex] there is a hole.

Substitute this value of x in the factored function to get the y value for the hole

We have
[tex]f(x) = \dfrac{x - 3}{x+2}[/tex]

[tex]\begin{aligned}f(2) &= \dfrac{-1 - 3}{-1 + 2}\\& = \dfrac{-4}{1}\\&= -4\end{aligned}[/tex]

So the hole occurs at   [tex](-1, -4)[/tex]

We can cοnclude that the variance frοm this randοm sample is nοt significantly different frοm 9.0 at a 5% level οf significance.

In mathematics, variance is a measure οf the spread οr dispersiοn οf a set οf data values. It is calculated as the average οf the squared differences frοm the mean. Variance is cοmmοnly used in statistics tο assess the variability οf a data set and is an impοrtant parameter in many statistical analyses.

Tο test whether the variance frοm this randοm sample is equal tο 9.0, we can use a chi-square test. The test statistic fοr this hypοthesis test is calculated as:

chi-square = (n - 1) × sample variance / pοpulatiοn variance

where n is the sample size.

Substituting the given values, we get:

chi-square = (9 - 1) × 8.01 / 9.0

chi-square ≈ 6.71

The critical chi-square value at a 5% level οf significance and 8 degrees οf freedοm (n - 1) is 15.51. Since the calculated chi-square value οf 6.71 is less than the critical chi-square value οf 15.51, we fail tο reject the null hypοthesis that the variance frοm this randοm sample is equal tο 9.0. In οther wοrds, there is nοt enοugh evidence tο suggest that the variance frοm this randοm sample is different frοm 9.0 at a 5% level οf significance.

Therefοre, we can cοnclude that the variance frοm this randοm sample is nοt significantly different frοm 9.0 at a 5% level οf significance.

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The value of the inverse function of f(x), that is, f⁻¹(33) is 2.

A function takes in values, applies specific operations to them, and produces an output. The inverse function acts, agrees with the outcome, and returns to the initial function. In general, switching the coordinates x and y is how an inverse is calculated. Although not strictly a function, this freshly constructed inverse is a relation.

To guarantee that the original function's inverse will likewise be a function, the original function must be a one-to-one function. Only when every second element matches the first value is a function deemed to be a one-to-one function.

Given that,

f(x)=4x³+1

Also, f⁻¹(33).

Thus, using the input and output corresponding values we have:

f(x) = 4x³ + 1 = 33

x³ = 8

Taking the cube root of both sides, we get:

x = 2

Hence, the value of the inverse function of f(x), that is, f⁻¹(33) is 2.

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Brian will receive £[tex]1.75[/tex] from Sue buying [tex]500[/tex] grams of berries.

The metric measure for a tiny amount of mass and weight is the gram (g). It weighs the same as one milliliter (or one cubic centimeter) of water. 1,000 grams make to one kilogram (kg). One liter liquid water weighs exactly one kilogram (kg).

Using a scale is the most accurate technique to measure in grams. Kitchen spoons and cups, for example, offer an approximate approximation. Moreover, always have a conversion tool or chart on available so you can estimate grams without a scale.

Brian paid a total of £[tex]15.75[/tex] for [tex]4\frac{1}{2}[/tex] kg of strawberries. To find out the price per gram, we can divide the total cost by the total weight:

Price per gram [tex]=[/tex]Total cost / Total weight

Total weight of strawberries [tex]= 4.5 kg = 4500 grams[/tex]

Price per gram [tex]=[/tex] £[tex]15.75 / 4500[/tex] grams

Price per gram [tex]=[/tex] £[tex]0.0035[/tex] per gram

So, Brian paid £[tex]0.0035[/tex] for every gram of strawberries he picked.

Cost of [tex]500[/tex] grams [tex]=[/tex] [tex]500 grams *[/tex] £[tex]0.0035/gram[/tex]

Cost of [tex]500[/tex] grams [tex]=[/tex] £[tex]1.75[/tex]

Therefore, Sue should pay Brian £[tex]1.75[/tex] for [tex]500[/tex] grams of strawberries.

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Answer:-225

Step-by-step explanation:The arithmetic series is given by:

a1 = -1, d=-2, n = 15

where a1 is the first term, d is the common difference, and n is the number of terms.

To evaluate this series, we can use the formula for the sum of an arithmetic series:

Sn = n/2 * [2a1 + (n-1)d]

where Sn is the sum of the first n terms.

Substituting the given values, we get:

S15 = 15/2 * [2(-1) + (15-1)(-2)]

= 15/2 * [-2 - 28]

= 15/2 * (-30)

= -225

Therefore, the sum of the first 15 terms of the arithmetic series with a1 = -1, d=-2 is -225.

Step-by-step explanation:

For a 151kg parcel, we can see that it falls under the category of "Over 100kg and up to 500kg", so the cost would be 21.10 Naira per kg:

Cost of sending 151kg parcel = 21.10 Naira/kg * 151 kg = 3187.10 Naira

For a 507kg parcel, we can see that it falls under the category of "Over 100kg and up to 500kg", so the cost would be 21.10 Naira per kg:

Cost of sending 507kg parcel = 21.10 Naira/kg * 507 kg = 10701.70 Naira

To find the cost of buying stamps, we need to check which category the number of stamps falls under. For 150 stamps, it falls under the category of "Over 10 stamps up to 50 stamps", so the cost would be 75.00 Naira:

Cost of buying 150 stamps = 75.00 Naira

For 213 stamps, it falls under the category of "Over 50 stamps up to 100 stamps", so the cost would be 105.00 Naira:

Cost of buying 213 stamps = 105.00 Naira

Answer:

7x+10

Step-by-step explanation:

2x(3x)+1x(x+10)

6x+1x+10

7x+10

In response to the query, we can state that As a result, the cubic equation factored form is: f(x) = (x+3)(x+1)(x-2) (x-2)

An equation is a mathematical statement that proves the equality of two expressions connected by an equal sign '='. For instance, 2x – 5 = 13. Expressions include 2x-5 and 13. '=' is the character that links the two expressions. A mathematical formula that has two algebraic expressions on either side of an equal sign (=) is known as an equation. It depicts the equivalency relationship between the left and right formulas. L.H.S. = R.H.S. (left side = right side) in any formula.

A cubic polynomial's graph is the one that is presented. We must determine the polynomial's x-intercepts or roots in order to determine its factored form.

The graph reveals the polynomial's three real roots, which are -3, -1, and 2.

As a result, the polynomial's factored form is:

f(x) = a(x+3)(x+1) (x-2)

where an is a constant that establishes the polynomial's leading coefficient. Using the graph's point (0,-6) as a reference, we may determine the value of a:

f(0) = a(0+3)(0+1)(0-2) = -6

If we condense this phrase, we get:

-6a = -6

When we multiply both sides by -6, we get:

a = 1

As a result, the cubic polynomial's factored form is:

f(x) = (x+3)(x+1)(x-2) (x-2)

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Answer:

Reflection only

The mirror line must pass through one of the sides of the triangle.

============================================================

Explanation:

The term "invariant" means "does not vary" i.e. "does not change".

An invariant point is fixed in place.

Here are the list of geometric transformations

Some geometry textbooks will mention "glide reflection", but that's really just a combination of translation and reflection. That means we can ignore it.

Let's go through each item to see if we can get two vertices to stay glued in their spot.  

In short we have eliminated: translations, rotations, and dilations. The only thing that works is reflections

Answer:

a. X ~ N(6.1, 1.2^2)

b. The median seedless watermelon weight is 6.1 kg.

c. The Z-score for a seedless watermelon weighing 7 kg is 0.75.

d. The probability that a randomly selected watermelon will weigh more than 5.1 kg is 0.7967.

e. The probability that a randomly selected seedless watermelon will weigh between 5.3 and 6 kg is 0.2454.

f. The 85th percentile for the weight of seedless watermelons is 7.2437 kg.

Step-by-step explanation:

a. X ~ N(6.1, 1.2^2)

b. The median of a normal distribution is equal to the mean, so the median seedless watermelon weight is 6.1 kg.

c. The Z-score for a seedless watermelon weighing 7 kg can be calculated as:

Z = (7 - 6.1) / 1.2 = 0.75

Therefore, the Z-score is 0.75.

d. To find the probability that a randomly selected watermelon will weigh more than 5.1 kg, we need to standardize the value using the formula:

Z = (X - μ) / σ

where X is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.

Z = (5.1 - 6.1) / 1.2 = -0.8333

Using a standard normal distribution table or a calculator, we can find the probability that Z is greater than -0.8333 to be 0.7967.

Therefore, the probability that a randomly selected watermelon will weigh more than 5.1 kg is 0.7967.

e. To find the probability that a randomly selected seedless watermelon will weigh between 5.3 and 6 kg, we need to standardize the values and find the area under the normal curve between the two Z-scores. The Z-scores for 5.3 kg and 6 kg are:

Z1 = (5.3 - 6.1) / 1.2 = -0.6667

Z2 = (6 - 6.1) / 1.2 = -0.0833

Using a standard normal distribution table or a calculator, we can find the probability that Z is between -0.6667 and -0.0833 to be 0.2454.

Therefore, the probability that a randomly selected seedless watermelon will weigh between 5.3 and 6 kg is 0.2454.

f. The 85th percentile for the weight of seedless watermelons can be found by finding the Z-score that corresponds to the 85th percentile of a standard normal distribution. Using a standard normal distribution table or a calculator, we can find the Z-score to be 1.0364.

To find the corresponding weight, we can use the formula:

Z = (X - μ) / σ

1.0364 = (X - 6.1) / 1.2

X = 7.2437

Therefore, the 85th percentile for the weight of seedless watermelons is 7.2437 kg.

Hope this helps! I'm sorry if it doesn't. If you need more help, ask me! :]

Answer:

a) N(6.1,1.2)
b) 6.1
c) 0.75
d) 0.7977   (or 79.77%)
e) 0.2143    (or 21.43%)
f) 7.3435 kg (using excel)    or      7.348 kg (using normal tables)

Step-by-step explanation:

a) normal distribution just need to be define we his mean and his standard deviation. You just need a sample higher than 30 or to be specified the population is normally distributed
X≈N(μ,σ)= N(6.1,1.2)

b) The median and mean are not necessarily the same, for a normal distribution, which is a symmetric distribution, those values are just the same


c) The equation for Z score is given by:

[tex]z=\frac{x-u}{\sigma}[/tex]

Replacing values:

[tex]Z=\frac{7-6.1}{1.2} =0.75[/tex]

d) first find the Zvalue for 5.1

[tex]Z=\frac{5.1-6.1}{1.2} =-0.83[/tex]

Now, find the probability from  Normal  Distribution table
P(Z>0.83) = 0.7977

e)

first find the Zvalue fo5.3 and 6

[tex]Z=\frac{5.3-6.1}{1.2} =-0.67[/tex]


[tex]Z=\frac{6-6.1}{1.2} =-0.08[/tex]

Now find probabilities from  Normal Distribution  table . Notice you need to subtract   P(Z>0.08)-P(Z>0.67) if you use a positive table.

P(-0.67<z<-0.08) = 0.2143


f) Find the Z score from a   Normal  Distribution table that give you an area of 0.8500 (or the closest value), im using excel for this one since the answer from tables have only TWO decimals and can be problematic if you need more decimal places.

this gives a Z of 1.036433389

now use the Zscore equation but solve for X

[tex]z\sigma+u=x[/tex]

x = 1.036433389(1.2)+6.1 = 7.3437   (Using excel)

If i use a table, the closes is 1.04 (0.8508 which is not exactly 0.85)

x = 1.04(1.2)+6.1= 7.348 (using table)

so be carefully with the input if the website needs the decimals places or not.

The value of x that maximizes the area of the figure is 1.5 inches, and the maximum area is 3.375 square inches.

A rectangle is a two-dimensional form that has four full sides and four right angles. Because the opposing sides are parallel and congruent, it is a unique kind of parallelogram.

To find the maximum area of the figure, we need to determine the value of x that will maximize the area.

The area of the figure can be calculated as:

A = (1.5x)(x) + (1/2)(1.5x)(6-2x)

Simplifying the expression:

[tex]A = 1.5x^2 + 4.5x - 3x^2[/tex]

[tex]A = -1.5x^2 + 4.5x[/tex]

To find the value of x that maximizes the area, we need to take the derivative of A with respect to x and set it equal to zero:

dA/dx = -3x + 4.5 = 0

Solving for x:

3x = 4.5

x = 1.5 inches

Now that we know x=1.5 inches maximizes the area, we can substitute it back into the area equation to find the maximum area:

[tex]A = -1.5(1.5)^2 + 4.5(1.5)[/tex]

A = 3.375 square inches

Therefore, the value of x that maximizes the area of the figure is 1.5 inches and the maximum area is 3.375 square inches.

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Answer:

If the question is x^2 + x - 20 = 0,

x = -4,5

However, if the question is x^2 - x - 20 = 0,

x = 4, -5

Step-by-step explanation:

WARNING: Your question is incorrect and so, I have tweaked it to make it correct

Completing the square of (x^2 + x - 20):

(x^2 - x) - 20

= (x^2 - x + (1/2)^2 - (1/2)^2) - 20

= (x^2 - x + (1/4))^2 - 1/4 - 20

= (x-1/2)^2 - 81/4

Hence,

(x-1/2)^2 - 81/4 = 0

(x-1/2)^2 = 81/4

x-1/2 = + or - 9/2

x = 9/2 + 1/2 (OR) x = -9/2 + 1/2

x = 5 (OR) x = -4

Hence,

x = -4,5 (CONSIDERING THE QUESTION IS x^2 + x - 20 = 0)

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The amount of money Leticia has after x time periods can be calculated using the formula:

f(x) = P(1 + r)^x

where P is the principal (initial amount invested), r is the annual interest rate (expressed as a decimal), and x is the number of time periods.

Substituting the given values, we get:

f(x) = 200(1 + 0.05)^x

Simplifying this expression, we get:

f(x) = 200(1.05)^x

Therefore, the exponential function that best represents this situation is:

f(x) = 200(1.05)^x

As a result, the compound inequality has the following solutions: m > 6 or m < -2 as by split both sides by -5 and reverse the inequality.

An inequality in mathematics is a comparison between two values or expressions that specifies whether they are equal, greater than, or less than one another. "!=" (greater than or equal to), and "=" (less than or equal to) are used to represent inequality (not equal to). For instance, the inequality "x > 2" denotes that the value of x is more than 2, while the inequality "5 7" denotes that 5 is less than 7. Variables may be utilized in inequalities, which can be used to indicate a variety of values or circumstances that satisfy the inequality.

given

Let's resolve each inequality in turn, then add the results:

4 - m < -2 (subtract 4 from both sides) (subtract 4 from both sides)

-m < -6 (divide both sides by -1 and invert the inequality) (divide both sides by -1 and reverse the inequality)

m > 6

12 < -5m + 2 (subtract 2 from both sides) (subtract 2 from both sides)

10 < -5m (split both sides by -5 and reverse the inequality) (divide both sides by -5 and reverse the inequality)-2 > m

As a result, the compound inequality has the following solutions: m > 6 or m < -2 as by split both sides by -5 and reverse the inequality.

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2xy - 2 = 0

Step-by-step explanation:

d2y/dx2 = 2xy - 2(1) = 0

The required possible dimensions of the​ rug are 11   by  [tex]\frac{r^2 - 8r - 33}{11}[/tex].

The area of the rectangular rug is given by the trinomial:

[tex]$r^2 - 8r - 33$[/tex]

To find the possible dimensions of the rug, we need to factor this trinomial. We can do this by finding two numbers that multiply to -33 and add up to -8. These numbers are -11 and 3:

[tex]$r^2 - 8r - 33 = (r - 11)(r + 3)$[/tex]

Therefore, the possible dimensions of the rug are:

[tex]$r - 11 = 0$[/tex] or [tex]$r + 3 = 0$[/tex]

[tex]$r = 11$[/tex]or $[tex]$r = -3$[/tex]

Since the dimensions of the rug cannot be negative, we can only take the positive value of r, which is r = 11.

Therefore, the possible dimensions of the rug are:

11   by  [tex]\frac{r^2 - 8r - 33}{11}[/tex]

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three time three equal 9

Answer:

Therefore, you would need to deposit $1,068.19 each month to have $700,000 in 35 years with a 7% interest rate.

Step-by-step explanation:

An exponential function is a function in which the independent variable x appears as an exponent.

a is the y-intercept.

b is the rate of change.

The function is exponential growth when b > 1.

The function is exponential growth when b < 1.

y-intercept is the initial value of the function when x is equal to 0.

Asymptote is the line that the function approaches but does not cross.

In Mathematics, an exponential function can be modeled by using this mathematical expression:

f(x) = ab^x

Where:

In Mathematics, an asymptote can be defined as a line which the graph of a given function approaches but would never touch or cross.

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The Answer to the questions Two ninths, Seven over twenty-Four, and Five Twelfths for understanding this we have to learn fractions.

Any number can be called a Fraction if it is in the form of [tex]\frac{p}{q}[/tex] where q≠0.

where p is the numerator of the fraction and q is the denominator of the fraction. The numerator can be 0 but the denominator never is zero.

The elements of a whole or entire set are expressed by fractions. A fraction consists of two components. The numerator is the figure at the top of the line. The denominator is the figure that occurs below the line.

The solutions to the questions are given below :

a. five-ninths minus two-sixths = Two-ninths

b. four-sixths  minus three-eighths  = Seven over twenty Four

c. four-sixths minus three-twelfths  = Five Twelfths

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The final function is f(x) =  ( [tex]b^3-4b^2[/tex]+[tex]63b[/tex]-487b)(b+7)+3021.

What is function?

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.

Here the given function f(x)=[tex]b^4 + 3b^3-35b^2-46b+28[/tex] and d(x)=b+7.

Now using long division then,

=>           [tex]b^3-4b^2[/tex]+[tex]63b[/tex]-487b

    [tex]\\b+7)\overline{b^4 + 3b^3-35b^2-46b+28 }\\\\[/tex]

    (-)      [tex]{b^4+7b^3}[/tex]

                  [tex]\overline{-4b^3+35b^2+46b-28}[/tex]

    (-)           [tex]-4b^3-28b^2[/tex]

                             [tex]\overline{63b^2-46b+28}[/tex]

     (-)                     [tex]63b^2+441b[/tex]

                                     [tex]\overline{-487b-28}[/tex]

     (-)                             [tex]\underline {-487b-3049}[/tex]

                                                3021

Quotient q(x) =   [tex]b^3-4b^2[/tex]+[tex]63b[/tex]-487b

Remainder r(x) = 3021

Hence the final function is f(x) =  ( [tex]b^3-4b^2[/tex]+[tex]63b[/tex]-487b)(b+7)+3021.

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The vertices of the image are given as follows:

X'(16.5, -9), Y'(13.5,-4.5) and Z'(4.5,-1.5).

The original vertices of the triangle are given as follows:

X(6, -2), Y(4,1) and Z(-2,3).

The vertices of the image are obtained applying the transformations in the context of the problem.

The first transformation is T(5,-4), which is a translation, meaning that 5 is added to the x-coordinate of each vertex, while 4 is subtracted from the y-coordinate of each vertex, hence:

X'(11, -6), Y'(9,-3) and Z'(3,-1).

The final transformation is the dilation with a scale factor of 1.5, meaning that each coordinate of each vertex is multiplied by 1.5, hence:

X'(16.5, -9), Y'(13.5,-4.5) and Z'(4.5,-1.5).

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[tex]\begin{array}{llll} \textit{Logarithm Cancellation Rules} \\\\ log_a a^x = x\qquad \qquad \stackrel{ \textit{we'll be using this one} }{a^{log_a (x)}=x} \end{array} \\\\[-0.35em] ~\dotfill\\\\ x^2\log(x)=10x^2\implies \log(x)=\cfrac{10x^2}{x^2}\implies \log(x)=10 \\\\\\ \log_{10}(x)=10\implies 10^{\log_{10}(x)}=10^{10}\implies x=10^{10}\implies x=10000000000[/tex]

When a variable (or set of letters) in an algebraic statement is substituted, its numerical value is used instead. The expression's total value can then be calculated.

Using the given functions, we have:

[tex]f(x) = x^2 + 3[/tex]

[tex]g(x) =[/tex] [tex]\sqrt{(x-4)}[/tex]

To find f(g(x)), we substitute g(x) for x in the expression for f(x):

[tex]f(g(x)) = (g(x))^2 + 3[/tex]

= ([tex]\sqrt{(x-4))^2 + 3}[/tex]

[tex]= (x-4) + 3[/tex]

[tex]= x - 1[/tex]

Therefore,[tex]f(g(x)) = x - 1.[/tex]

To find g(f(x)), we substitute f(x) for x in the expression for g(x):

[tex]g(f(x)) =[/tex] [tex]\sqrt{((f(x)) - 4)}[/tex]

= [tex]\sqrt{((x^2 + 3) - 4)}[/tex]

= [tex]\sqrt{(x^2 - 1)}[/tex]

= |x|[tex]\sqrt{(x^2 - 1)}[/tex]

Therefore,  [tex]g(f(x)) =[/tex] |x|  [tex]\sqrt{(x^2 - 1)}[/tex]

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Answer:  Large box weight 40 pounds, and small box weight 30 pounds.

Step-by-step explanation:

Let L be the weight of a large box, and S be the weight of a small box.

L + S = 70

50L + 60S = 3800

We change the first equation into the form of L, and we get:

L = 70 - S

And we plugin L = 70 - S into the second equation.

  50(70 - S) + 60S = 3800

3500 - 50S + 60S = 3800

          3500 + 10S = 3800

         -3500             -3500

                       10S = 300

                          S = 30

 L + S = 70

L + 30 = 70

  - 30     - 30

       L = 40

[tex]csc(-x)-1~~ = ~~\cfrac{cos^2(x)}{csc(-x)+1} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{cos^2(x)}{csc(-x)+1}\implies \cfrac{1-sin^2(x)}{\frac{1}{sin(-x)+1}}\implies \cfrac{1-sin^2(x)}{\frac{1}{-sin(x)+1}}\implies \cfrac{[1-sin(x)][1+sin(x)]}{ ~~ \frac{-1+sin(x)}{sin(x)} ~~ } \\\\\\\ [1-sin(x)][1+sin(x)]\cdot \cfrac{sin(x)}{-1+sin(x)} \\\\\\\ [1-sin(x)][1+sin(x)]\cdot \cfrac{sin(x)}{-[1-sin(x)]} \\\\\\\ [1+sin(x)][-sin(x)] \implies \boxed{-sin(x)-sin^2(x)} \\\\[-0.35em] ~\dotfill[/tex]

[tex]csc(-x)-1\implies \cfrac{1}{sin(-x)}-1\implies \cfrac{1}{-sin(x)}-1\implies \boxed{\cfrac{-1-sin(x)}{sin(x)}}[/tex]

now, as you can see is a no dice, before doing many simplifications, I often check them graphically, so I know what's cooking, anyhow, Check the picture below, they are definitely not the same, the red one is the one on the right-hand-side.

The equation for the hyperbola is [tex]y^{2} -4x^{2} =144[/tex]

A hyperbola is a type of conic section, formed by the intersection of a plane with two cones that have a common vertex. It consists of two separate curves that are mirror images of each other, each of which has two branches that extend to infinity.

By Given figure,

C = (0, 0)

V (0, ±6)

b = 6

equation of asymptotes;  [tex]y=[/tex] ±[tex]\frac{a}{b}x[/tex]

then, y = ± [tex]\frac{a}{6} * x[/tex]

given the asymptotes, y = ±2x

by comparing we get, a/6 = 2

a = 12

We know the equation of hyperbola,   [tex]\frac{y^{2} }{a^{2} } -\frac{x^{2} }{b^{2} } =1[/tex]

putting the values of a and b,     [tex]\frac{y^{2} }{12^{2} } -\frac{x^{2} }{6^{2} } =1[/tex]

[tex]\frac{y^{2} }{144} } -\frac{x^{2} }{36} } =1[/tex]

After Simplification, [tex]y^{2} -4x^{2} =144[/tex]

Therefore, the equation for the hyperbola is [tex]y^{2} -4x^{2} =144[/tex]

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