5. Explain the properties of logarithms that you
would need to use to solve the following:
log₂2x + log₂4 = 5

The distance the ladder slipped down is 21cm.

The ladder, wall and the base of the wall form a right triangle. The ladder is the hypotenuse. The wall is the height and the distance of the foot of the ladder to the base of the wall is the base.

Pythagoras theorem would be used to determine the length of the wall.

The Pythagoras theorem: a² + b² = c²

where:

100 cm = 1 meter

3 x 100 = 300 cm

a²  + 50²  = 300²

a² + 2500  = 90,000

a² = 90,000 - 2500

a²  = 87,500

a = √87500

a = 296 cm

Height when the ladder slipped down:

(70 + 50)²  + a²  = 300²

120²  + a² = 300²

14,400 + a² = 90,000

a² = 90,000 - 14,400

a²  = 75,600

a = √75,600

a = 275

Distance the ladder slipped = 296- 275 = 21 cm

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Answer: 1. Slope equals 3/4 or 0.75,  2. Slope equals -5/2 or -2.5

Step-by-step explanation:

1. Subtract y coordinates 4 and -2 to get 6. Then subtract x coordinates 4 and -4 to get 8, so that is equal to 6/8. Then simplify to get 3/4 or 0.75

2. Subtract y coordinates 10 and -5 to get -15. Then subtract x coordinates 0 and -6 to get 6, so that is equal to-15/6. Then simplify to get -5/2 or -2.5.

The approximate speed of the car when the skid marks are 20 feet long is 8.944 miles per hour.

A formula is an expression in mathematics that denotes a connection or rule. In mathematics, science, and engineering, numbers and variables are described and computed using formulas. They can be stated verbally or symbolically, such as in verbal descriptions, or mathematically or symbolically, such in algebraic equations. Formulas can be applied to evaluate data, solve issues, and make predictions. They are utilised in a variety of disciplines, including physics, chemistry, biology, finance, and statistics, and are crucial for comprehending and explaining the physical world.

The skidding distance is given by the formula:

d = 2 * √(5x)

Isolating the value of x we have:

x = (d / (2 * √(5)))

Substituting the value of d = 20 we have:

x = (20 / (2 * √(5))) ≈ 8.944 miles per hour

Hence, the approximate speed of the car when the skid marks are 20 feet long is 8.944 miles per hour.

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a. Construct a 90% confidence interval estimate for the mean content viewing time per week in this area. Isusa

90% confidence interval estimate for the mean content viewing time per week: (41.75, 46.25)

b. Construct a 90% confidence interval estimate for the population proportion of residents who have HD on at least one television set. Osas

90% confidence interval estimate for the population proportion: (0.1867, 0.3833)

c. Suppose that the market researcher wants to take another survey in a different location. What sample size is required to be 90% confident of estimating the population mean content viewing time to within + 2 hours assuming that the population standard deviation is equal to 6 hours? (Type a whole number.)

Sample size required: 653

d. How many respondents need to be selected to be 90% confident of being within 10.05 of the population proportion who have HD on at least one television set if no previous estimate is available? (Type a whole number.) Enter your answer in each of the answer boxes.

Sample size required: 461

e. Based on (C) and (d), how many respondents should the market researcher select if a single survey is being conducted? (Type a whole number.) Enter your answer in each of the answer boxes.

Sample size required: 653

Answer:

y = x - 2

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

calculate m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (0, - 2) and (x₂, y₂ ) = (2, 0) ← 2 points on the line

m = [tex]\frac{0-(-2)}{2-0}[/tex] = [tex]\frac{0+2}{2}[/tex] = [tex]\frac{2}{2}[/tex] = 1

the line crosses the y- axis at (0, - 2 ) ⇒ c = - 2

y = x - 2 ← equation of line

Therefοre , the sοlutiοn οf the given prοblem οf area cοmes οut tο be The cοne hοlds 43.96 in³mοre ice cream than the cup, sο that is the cοrrect respοnse.

Calculating hοw much space wοuld be needed tο fully cοver the οutside will reveal its οverall size. When determining the surface οf such a trapezοidal fοrm, the surrοundings are additiοnally taken intο accοunt. The surface area οf sοmething determines its οverall dimensiοns. The number οf edges here between cubοid's fοur trapezοidal extremities determines hοw much water it can hοld inside.

Here,

We must determine the vοlumes οf the cοne and the cup in οrder tο cοmpare hοw much ice cream each cοntainer can carry.

=> Cοne vοlume =  (1/3)r²h.

=>  Cup vοlume = (1/4)d²h

where r is the cοne's radius (w/2), h is the cοne's height (x), d is the cup's width (y), and z is the cup's height.

Inputting the numbers prοvided yields:

=>  Cοne vοlume =   (1/3)π(2²)(6) = 16π cubic inches

=>  Cup vοlume =  (1/4)π(6²)(2) = 9π cubic inches

As a result, the cοne has a larger capacity than the cup.

The cοne hοlds 43.96 in³mοre ice cream than the cup, sο that is the cοrrect respοnse.

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The solution to the expression (5x² - 6x) / 3x is (5/3)x - 2

An equation is an expression that shows the relationship between numbers and variables using mathematical operators such as addition, subtraction, exponent, multiplication and division.

Equations can be classified based on degree as linear, quadratic, cubic and so on.

Given the polynomial:

5x² - 6x

Dividing the polynomial by the monomial 3x:

= (5x² - 6x) / 3x

= [x(5x - 6)] / 3x

= (5x - 6) / 3

= (5/3)x - 2

The solution to the equation is (5/3)x - 2

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Answer:

sin

cos

tan

csc

sec

cot

Step-by-step explanation:

easy to remember that they are reciprocals of each other

Answer:

x=7 & y=-7

Step-by-step explanation:

from the 2nd equation 7x+7y=0 we can get that

x=-y

substitute that into the first equation to get

3y-5y=14

-2y=14

y=-7

then

x=7

The angle of depression from the sun that makes the shadow is 69.1°.

The angle of depression from the sun can be calculated using the formula, tan θ = h/l, where θ is the angle of depression, h is the height of the building, and l is the length of the shadow. In this case, h is 400ft, and l is 75ft. Plugging these numbers into the formula gives us tan θ = 400/75, which simplifies to [tex]θ = tan^-1(400/75)[/tex]. Using a calculator, this gives us an angle of depression of θ = 69.1°. Therefore, the angle of depression from the sun that makes the shadow is 69.1°.

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We have drawn the number line and included the given points, see attached number line.  

What is the fraction?

A fraction is used to represent the portion/part of the whole thing. It represents the equal parts of the whole. A fraction has two parts, namely the numerator, and denominator. The number on the top is called the numerator, and the number on the bottom is called the denominator.

On the number line above, the decimal values and fractions are labeled as follows:

0.63 is labeled between 0 and 1.

34/100 is labeled at the same point as 0.34, between 0 and 1.

0.02 is labeled between 0 and 1.

7/10 is labeled at the same point as 0.7, between 0 and 1.

Therefore, we have drawn the number line and included the given points, see attached number line.  

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The average daily balance is $1,000, the finance charge is $14.7, and the new balance is $3,029.4.

To calculate the average daily balance, finance charge, and new balance using the average daily balance method, follow the below steps:

Step 1: Obtain the balance owed for each day. To do so, determine the balance at the beginning of each day and the amount charged or credited during the day.

Step 2: Add up the balances for each day during the billing cycle, then divide by the number of days in the billing cycle. This calculation results in the average daily balance.

Step 3: Multiply the average daily balance by the daily interest rate to calculate the finance charge.Step 4: Add the finance charge to the previous balance to get the new balance. Example: Suppose an account has the following daily balances and APR, calculate the average daily balance, finance charge, and new balance using the average daily balance method:

Day 1 balance: $1,000Day 2 balance: $1,200Day 3 balance: $800Day 4 balance: $1,000APR: 18%The number of days in the billing cycle is 30.Average daily balance:($1,000 + $1,200 + $800 + $1,000) ÷ 4 = $1,000Finance charge:$1,000 × 0.18 ÷ 365 = $0.49 per day.$0.49 × 30 = $14.7New balance: Previous balance + finance charge = $3,014.7 + $14.7 = $3,029.4

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Answer:

[tex]y = -\dfrac{1}{4}x + 1[/tex]

Step-by-step explanation:

Slope of the line can be found by taking two points on the line, (x1, y1) and (x2, y2)

Find the difference in y coordinates, y2 - y1  (called rise)

Find the difference in corresponding x coordinates x2 - x1 (called run)

rise/run gives the slope

Two distinguishable points on the line graph are (0, 1) and (4, 0)

rise = 0 - 1 = - 1
run = 4 - 0 = 4

Slope = -1/4

The equation of the line in general form is

y = mx + b

where

m = slope

b = y-intercept (value of y when x = 0)

Here m = -1/4

b = 1

So equation of the line is

[tex]y = -\dfrac{1}{4}x + 1[/tex]

The only value of x for which (-9)(x+4) = 0 is x = -4.

Calculations that include changeable altering multipliers, joining, removal, and random subdivision must be done. They could accomplish the following if they united: An algorithm, some data, and a mathematical problem.

The expression (-9)(x+4) can be simplified by using the distributive property of multiplication:

(-9)(x+4) = -9x - 36

To find the values of x for which (-9)(x+4) = 0, we can set the expression equal to zero and solve for x:

-9x - 36 = 0

Adding 36 to both sides gives:

-9x = 36

Dividing both sides by -9 gives:

x = -4

Therefore, the only value of x for which (-9)(x+4) = 0 is x = -4.

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a .The probability that x of the top 25 pairs play east-west is: P(X = x) = (25 choose x) / (50 choose 25) , b. The probability that the top 5 pairs all play in the same direction is: P(all 5 play the same direction) = (1/2) * (24/49) * (23/48) * (22/47) * (21/46), c. The pmf of X is given by P(X = x) = [(n choose x) * (n choose n - x)] / [(2n choose n)], and the expected value and variance of X are E(X) = n/2 and V(X) = (n² - 1)/12.

P(X = x) = [(n choose x) * (n choose n - x)] / [(2n choose n)] yields the pmf of X. , and X's expected value and variance are, respectively, E(X) = n/2 and V(X) = (n² - 1)/12.

(a) Let X be the number of top 25 pairs that play east-west. Since there are 25 east-west pairs and 25 north-south pairs, the total number of ways to choose 25 pairs to play east-west is (50 choose 25) = 50! / (25! * 25!). Similarly, the total number of ways to choose x top pairs to play east-west is (25 choose x) * (25 choose 25 - x) = (25 choose x), since the remaining 25 pairs automatically play east-west. Therefore, the probability that x of the top 25 pairs play east-west is:

P(X = x) = (25 choose x) / (50 choose 25)

(b) The probability that the top pair plays east-west is 25/50 = 1/2. Once the top pair has been assigned a direction, the probability that the second pair plays in the same direction is 24/49, the probability that the third pair plays in the same direction is 23/48, and so on. Therefore, the probability that the top 5 pairs all play in the same direction is:

P(all 5 play the same direction) = (1/2) * (24/49) * (23/48) * (22/47) * (21/46)

(c) Let X be the number of top n pairs that play east-west, where there are 2n pairs total. Then X can take on values from 0 to n. To find the pmf of X, we can use the same method as in part (a):

P(X = x) = [(2n - n) choose x] / [(2n) choose n]

= [(n choose x) * (n choose n - x)] / [(2n choose n)]

The expected value of X is:

E(X) = sum[x * P(X = x), {x, 0, n}]

= sum[x * (n choose x) * (n choose n - x) / (2n choose n), {x, 0, n}]

Using the identity sum[x * (n choose x) * (n choose k - x), {x, 0, k}] = k * (n choose k), we can simplify this to:

E(X) = n/2

The variance of X is:

V(X) = E(X²) - E(X)²

= sum[x² * P(X = x), {x, 0, n}] - (n/2)²

= sum[x * (n choose x) * (n choose n - x) / (2n choose n), {x, 0, n}] - (n/2)²

= (n² - 1)/12

Therefore, the pmf of X is given by P(X = x) = [(n choose x) * (n choose n - x)] / [(2n choose n)], and the expected value and variance of X are E(X) = n/2 and V(X) = (n² - 1)/12, respectively.

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The surface area of the three-dimensional figure represented by this net is 54 square inches, which is calculated by multiplying the area of the base (9 square inches) by the height of the four congruent triangles (8 inches).

The surface area of the three-dimensional figure represented by this net is 54 square inches. To calculate this, first we need to find the area of the base. The base is a square, with side lengths of 3 inches each. Therefore, the area of the base is 3 x 3 = 9 square inches. Next, we need to find the height of the four congruent triangles. The height is 8 inches. Finally, we can calculate the surface area by multiplying the area of the base (9 square inches) by the height of the four congruent triangles (8 inches). This gives us a total surface area of 54 square inches.

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Answer:

50%

Step-by-step explanation:

it's just half 80 so it's 50%

Step-by-step explanation:

To solve this problem, we can use the formula for pressure:

pressure = force / area

We are given the pressure and the force, so we can rearrange the formula to solve for the area:

area = force / pressure

Substituting the values we have:

area = 180 N / 200 N/m²

area = 0.9 m² (to 1 decimal place)

Therefore, the area of the base of the box is 0.9 square meters.

The number of triangles that can be formed from a common vertex on the polygon shown is three triangles.

To form triangles on polygons, you can draw diagonals connecting the vertices of the polygon. A diagonal is a line segment that connects two non-adjacent vertices of a polygon. By drawing diagonals, you can create triangles within the polygon.

From a hexagon, we can draw three diagonals from each vertex. So, from a common vertex, we can draw three diagonals and form three triangles. Since there are six vertices in a hexagon, we can choose any one of these vertices as the common vertex and form three triangles.

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Answer:

8.2km

Step-by-step explanation:

North and East are perpendicular directions;

If you illustrate the distances covered in each direction and join the two lines, it will form a right-angle triangle;

This means pythagoras applies:

(8)² + (2)² = 64 + 4 = 68

The distance apart is, therefore:

[tex] \sqrt{68} = 8.2462112512 [/tex]

Yes, there is direct relatiοn in amοunt οf dry dοg fοοd and the amοunt οf wet dοg fοοd.

If amοunt οf dry dοg fοοd  is x and the amοunt οf wet dοg fοοd is y then the relatiοn is given as: x = 3y

Tο determine whether the relatiοnship between the amοunt οf dry dοg fοοd and the amοunt οf wet dοg fοοd is prοpοrtiοnal, we need tο check whether the ratiο οf the twο quantities remains cοnstant.

A) yes, there is direct relatiοn in amοunt οf dry dοg fοοd and the amοunt οf wet dοg fοοd. as amοunt οf dry dοg fοοd increases there alsο increase in amοunt οf wet gοοd.

if amοunt οf dry dοg fοοd  is x and the amοunt οf wet dοg fοοd is y then the relatiοn is given as:

x = 3y

fοr example lets take values frοm first rοw

4.5 = 3 × 1.5

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Complete question:

A scientist recorded the duration of the eruptions of the Old Faithful geyser in Yellowstone National Park that occurred during a one-month time period. The histogram below shows the distribution of the duration, in seconds, of the eruptions. By analyzing this data, the scientist can gain a better understanding of the behavior of the Old Faithful geyser and potentially make predictions about future eruptions.

The histogram shows that the majority of the eruptions had a duration between 120 and 140 seconds. There were also a significant number of eruptions with a duration between 100 and 120 seconds, and between 140 and 160 seconds.

It can also help them identify any unusual eruptions that may have a significantly shorter or longer duration than the majority of the eruptions
This data can help the scientist understand the typical duration of eruptions at the Old Faithful geyser in Yellowstone National Park..

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Answer:

B

Step-by-step explanation:

The only way for two numbers to = zero when they're added together is if one is negative. There are no two positive numbers that when added together will equal 0, therefore if p and t are positive, their sum cannot be 0

Answer:

  99.6 cm²

Step-by-step explanation:

You want to know the area of a sector that has radius 6 cm, and subtends an arc of 317°.

The formula for the area of a sector is ...

  A = 1/2r²θ . . . . . . where θ is the central angle in radians

  A = 1/2(6 cm)²(317·π/180) ≈ 99.6 cm²

The area of the sector is about 99.6 square centimeters.

The difference in commission for selling goods worth Rs. 20 lakhs and Rs. 30 lakhs is Rs. 7,500.

The commission on sales worth Rs. 20 lakhs would be:

1% of (20 lakhs) = 0.01 x 20,00,000 = Rs. 20,000

The commission on sales worth Rs. 30 lakhs would be:

1% of (25 lakhs) + 1.5% of (5 lakhs) = (0.01 x 25,00,000) + (0.015 x 5,00,000) = Rs. 27,500

The difference in commission between the two sales would be:

Rs. 27,500 - Rs. 20,000 = Rs. 7,500

Therefore, the difference in commission for selling goods worth Rs. 20 lakhs and Rs. 30 lakhs is Rs. 7,500.

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The given ordered pair is not a solution of the linear system.

Linear equations are mathematical equations that describe the relationship between two or more variables in a straight line. They can be written in the form of:

y = mx + b

To check if the ordered pair (-2, -3) is a solution to the system of linear equations, we need to substitute x = -2 and y = -3 into both equations and see if they are satisfied.

For the first equation y = 3x + 3, we have:

-3 = 3(-2) + 3

-3 = -3

This is true, so (-2, -3) satisfies the first equation.

For the second equation y = -x - 5, we have:

-3 = -(-2) - 5

-3 = 3

This is false, so (-2, -3) does not satisfy the second equation.

Since (-2, -3) satisfies the first equation but not the second equation, it is not a solution to the system of linear equations. Therefore, the answer is no.

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Therefore , the solution of the given problem of function comes out to be  f(N) = 360/N the cost per individual for any number of contributors to the gift is provided by this function.

The math lesson includes a wide range of topics, such as mathematics, integers, as well as one's divisions, architecture, and both actual mostly and fictitious geographic locations. The relationships between various components that all cooperate to create the same outcome are covered by a work. A utility is composed of numerous unique elements that work together to produce unique outcomes for each input.

Here,

To figure out how much each individual would spend, we need to divide the cost of the gift by the number of people contributing. Here are the estimates for various populations:

Each person would spend $180 ($360 / 2) if there were two people splitting the expense.

The expense would be divided among three people, who would each pay

=>  $120 = ($360 / 3).

If the cost were divided among five individuals, each person would pay =>  $72 = ($360 / 5).

Ten people would split the expense equally, each spending

=>  $36 =  ($360 / 10).

Each person would spend

=>  $3.60 = ($360 / 100)

if there were 100 people splitting the expense.

II. The algorithm is as follows:

Cost per individual equals total expense / number of individuals

=> f(N) = C/N

where f(N) is the price per individual.

The function for the cost per individual, for instance, is:

=>  f(N) = 360/N

The cost per individual for any number of contributors to the gift is provided by this function.

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Using the trigonometry function, the value of y = 3√2.

The area οf mathematics knοwn as trigοnοmetry is cοncerned with analyzing the relatiοnships between the sides and angles οf a right-angled triangle. A trigοnοmetric ratiο called the sides' ratiο serves as a representatiοn οf the relatiοnship. Trigοnοmetric ratiοs cοme in six varieties: sine, cοsine, tangent, cοtangent, secant, and cοsecant.

Trigοnοmetric functiοns are used tο calculate the lengths and angles οf geοmetric fοrms frοm knοwn οr measured angles. As a result οf the need tο cοmpute angles and distances in numerοus fields, including astrοnοmy, mapmaking, surveying, and artillery range finding, trigοnοmetry was develοped.

Since it is a 90-60-30 triangle, y is s √2

s = 3

so, y = 3√2

y = 3√2

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Answer: They are not similar.

Step-by-step explanation: The angle measures are not the same for both triangles which makes them not similar to each other.