Write as a logarithm with a base of 4. -4

Answer: 601 meters

Step-by-step explanation:

First, let's find the midpoint between the "Falcon Flyer" and the "Sprinter".

The x-coordinate of the midpoint is (50 + 400)/2 = 225.

The y-coordinate of the midpoint is (300 + 550)/2 = 425.

So the midpoint is (225, 425).

Now we need to find the distance between the midpoint and the "Air Rider". Using the distance formula:

d = sqrt[(800 - 225)^2 + (250 - 425)^2]

d = sqrt[(575)^2 + (-175)^2]

d = sqrt[330625 + 30625]

d = sqrt(361250)

d ≈ 601.041

So the distance you will walk on this shortcut is approximately 601 meters.

Volume First Cylinder: 50.24 in³

Volume Second Cylinder: 127.5625 inch³

Volume Third Cylinder: 904.32 inch³

Volume First Cylinder:

= πr²h

= 3.14(2)² (4)

= 3.14 x 16

= 50.24 in³

Volume Second Cylinder:

= πr²h

= 3.14(2.5)² (6.5)

=3.14 (6.25)(6.5)

= 127.5625 inch³

Volume Third Cylinder:

= πr²h

= 3.14(6)² (8)

=3.14 (36)(8)

= 904.32 inch³

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There is a 0.9830, or 98.30%, probability that the mean clock life will exceed 12.5 years.

Probability is a metric used to express the possibility or chance that a particular event will occur.

Probabilities can be expressed as fractions from 0 to 1, as well as percentages from 0% to 100%.

Probability is simply the possibility that something will happen. When we don't know how something will turn out, we can talk about the possibility of one outcome or the likelihood of several.

The study of events that fit into a probability distribution is known as statistics.

So, the likelihood that the average clock life would exceed 12.5 years:

This is the p-value of Z when X = 12.5 subtracted by 1.

So,

Z = X - μ/σ

Z = 12.5 - 14/.07071

Z = -2.12

Z = The pvalue for -2.12 is 0.0170.

1 - 0.0170 = 0.9830

The likelihood that the mean clock life will be longer than 12.5 years is 0.9830, or 98.30%.

Therefore, there is a 0.9830, or 98.30%, probability that the mean clock life will exceed 12.5 years.

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Complete question:

A production manager at a wall clock company wants to test their new wall clocks. The designer claims they have a mean life of 14 years with a variance of 25. If the claim is true, in a sample of 50 wall clocks, what is the probability that the mean clock life would be greater than 12.5 years? Round your answer to four decimal places.

Thus, the value of b over a for the given logarithmic function is obtained as: b/a = 1 + √2.

The logarithm is exponentiation's opposite function in mathematics. This indicates that the exponent with which b must be raised in order to obtain a number x is the logarithm of x to the base b.

A base must still be raised to a certain exponent or power, or logarithm, in order to produce a specific number. If bˣ = n, then x is the logarithm of n to the base b, which is expressed mathematically as x = logb n.

Given that:

log9(a) = log15(b) = log25(a + 2b)

Let ,  log9(a) = log15(b) = log25(a + 2b) = x

Then,

log9(a) = x (taking antilog)

log3²(a) = x

a = 3²ˣ

log15(b) = x  (taking antilog)

log(3*5)(b) = x

b = 3ˣ.5ˣ

log25(a + 2b) = x   (taking antilog)

log5²(a + 2b) = x

a + 2b = 5²ˣ

Now,

b²/a = a + 2b

(b/a)² = 1 + 2*(b/a)

If t = b/a, then t² - 2t -1 = 0

On factorization:

t = 1 ± √2 ( a and b are positive integer)

b/a = 1 + √2

Thus, the value of b over a for the given logarithmic function is obtained as: b/a = 1 + √2.

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The correct option is C. mean, that is Mr. Greene should use the mean temperature, which is 77.86°F.

In mathematics, the mean value is a measure of central tendency of a set of numbers. It is also known as the arithmetic mean and is found by adding all the numbers in a set and dividing the sum by the total number of values. The mean is commonly used to represent the typical or average value of a set of data. It is a useful tool in statistics and other branches of mathematics for analyzing data and making predictions. The mean value can be affected by outliers or extreme values in a set of data, which can skew the results. However, in many cases, the mean provides a reliable and useful measure of central tendency.

Mr. Greene should use the mean temperature, which is 77.86°F, to summarize the temperatures for each day. The mean is a good measure of central tendency when the data set is approximately symmetric and has no significant outliers, which appears to be the case here. Additionally, using the mean will provide a single number that is representative of the entire data set. The interquartile range and mean absolute deviation are measures of spread and do not provide a summary of the temperatures for each day, and the maximum temperature is just one value and may not be representative of the entire data set.

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1. the APR for this loan is 6%.

2.  the unearned interest is $118.83.

3.  the monthly payment necessary to amortize the $80,000 mortgage at 6% annual interest for 25 years is $506.69.

APR = (finance charge / loan amount) / number of years * 100%

In this scenario, the loan amount is $2400 and the finance charge is $288. Since it's a two-year loan, the number of years is 2

APR = (288 / 2400) / 2 * 100%

APR = 0.06 * 100%

APR = 6%

b.

Number of payments = 4 years * 12 months/year = 48

If the loan is paid off in 3 years instead of 4 years, the number of payments made is:

Number of payments made = 3 years * 12 months/year = 36

Substituting these values into the formula, we get:

Unearned interest = (1426 / 48) * (48 - 36)

Unearned interest = $118.83

3.

To find the monthly payment necessary to amortize a mortgage, we need to use the formula:

M = P * (r / (1 - (1 + r)^(-n)))

where M is the monthly payment, P is the principal (loan amount), r is the monthly interest rate, and n is the total number of payments (in months).

In this scenario, the loan amount is $80,000, the annual interest rate is 6%, and the loan term is 25 years. To find the monthly interest rate, we divide the annual interest rate by 12 (since there are 12 months in a year):

Monthly interest rate = 6% / 12 = 0.005

To find the total number of payments, we multiply the number of years by 12:

Number of payments = 25 years * 12 months/year = 300

Substituting these values into the formula, we get:

M = 80000 * (0.005 / (1 - (1 + 0.005)^(-300)))

M ≈ $506.69

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Rates of first and second mechanics are 75 and 60 per hour respectively.

A mathematical statement or relation between two or more numeric values, variable and mathematical operations via equal sign is called an Equation.

Elimination method is a kind of method to solve simultaneous equations in two variables. In elimination method we compare coefficient of one of two variables and in next step we eliminate that variable using addition or subtraction and form another relation with only one variable and find the value for that one variable and then substituting that value initial equation we can get the value of another variable.

Let the rate of first mechanic and second mechanic be x and y per hour respectively.

Sum of the two rates is 135 per hour. So the suitable equation will be,

x+y = 135 ............... (i)

Since they charged together 1650 after working 10 hours by first and 15 hours by second mechanics. So the suitable equation this time will be,

10x+15y = 1650

5(2x+3y) = 1650

2x+3y = 1650/5

2x+3y = 330 ............... (ii)

Multiplying 2 with equation (i) and then subtracting it from equation (ii) we get,

(2x+3y) - 2(x+y) = 330 - 2*135

2x+3y-2x-2y = 330-270

y = 60

Substituting y=60 in equation (i) we get,

x+60 = 135

x = 135-60 = 75

Hence, the rates charged per hour are 75 and 60 per hour respectively.

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The possible values of a are 3 and -3.

We know that the coefficient of x⁷ in the given expression is 1215, the expression is:

[tex]x^3(ax + 3)^5[/tex]

The term with x⁷ will be:

[tex]5*(3x^7*a^4) = (15a^4)*x^7[/tex]

Then we need to solve the equation:

15a⁴ = 1215

a⁴ = 1215/15

a⁴ = 81

a = ⁴√81 = ± 3

These are the possible values of a.

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the correct response to the question is: Yes, No, Yes, No and Yes.

We can determine if each set of numbers can be the lengths of the sides of a right triangle by evaluating each set.

Evaluating each set :

#1 :

5, 12, 13

5² + 12² = 13²

25 + 144 = 169

169 = 169

It is a Yes

#2 :

12, 35, 20

12² + 35² = (20√3)²

144 + 1225 = 400*3

1369 ≠ 1200

It is a No

#3 :

5, 10, 5√5

5² + 10² = (5√5)²

25 + 100 = 25(5)

125 = 125

It is a Yes

#4 :

8, 12, 15

8² + 12² = 15²

64 + 144 = 225

208 ≠ 225

It is a No

#5 :

20, 99, 101

20² + 99² = 101²

400 + 9801 = 10201

10201 = 10201

It is a Yes

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The question lacks some details, see complete question in the attached image.

The length of AB can be found to be 9. 85 units .

To find the length of AB, we can use the distance formula which is:

d = √ ( ( x2 - x1) ^ 2 + ( y2 - y1) ^2 )

The vertices for AB are:

A - ( 1, 7 )

B - ( 10, 3 )

When the values are plugged into the formula, we get :

d = √ ( ( x2 - x1) ^ 2 + ( y2 - y1) ^2 )

d = √ ( ( 10 - 1 )^ 2 + ( 3  - 7) ^2 )

d = √ ( 81 + 16 )

d = √ 97

d = 9. 85 units

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The ice rink is 14.42 units away from the origin and other solutions are added below

First, we have the library at (7,8), which is 7 units to the right of the origin and 8 units up from the origin.

Next, we have the market at (3,6), which is 4 units to the left of the library (7-4=3) and 2 units down from the library (8-2=6).

Then, we have the bank at (3,8), which has the same x-coordinate as the market (3) and a y-coordinate of 8.

We have the school at (10,8), which has the same y-coordinate as the bank (8) and an x-coordinate of 10.

The pool is located at (7,3), which is 5 units down from the school (8-5=3) and 3 units to the left from the school (10-3=7).

Finally, the park is located at (7,1), which has the same x-coordinate as the pool (7) and a y-coordinate of 1.

The completed table is:

Point Ordered Pair (x, y)

Library (7,8)

Market (3,6)

Bank (3,8)

School (10,8)

Pool (7,3)

Park (7,1)

Bao's new location can be found by moving 5 blocks down from his current location of (3,13) and then 2 blocks right.

This gives us the new coordinates of (5,8) for Boo.

The new location is 5 units right and 8 units up of the origin

To get to the baseball diamond, he traveled 3 blocks to the right from his starting point (3 - 3 = 0) and 8 blocks down (15 - 8 = 7), which gives us the new coordinates of (3, 15) for the baseball diamond.

Bao's starting point was (0, 7).

Bao's starting point can be found by moving 5 blocks to the left from his current location of (14, 13) and then 10 blocks down.

This gives us the new coordinates of (9,3) for Bao's starting point.

The distance between the ice rink and the origin can be found using the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

In this case, we have:

d = √((12 - 0)^2 + (8 - 0)^2) = √(144 + 64) = √(208) ≈ 14.42

So the ice rink is approximately 14.42 units away from the origin.

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The length of the rectangle is 10 inches and the width is 4 inches.

What is the area of the rectangle?

To find the area of a rectangle, we multiply the length of the rectangle by the width of the rectangle.

Let's assume that the width of the rectangle is "x" inches.

From the given information, we can write an equation for the length "L" in terms of the width "x":

L = 2x + 2

We also know that the area of the rectangle is 40 square inches:

A = L * x = 40

Substituting the expression for "L" in terms of "x" into the area equation, we get:

(2x + 2) * x = 40

Expanding the left-hand side of the equation and simplifying, we get:

2x² + 2x - 40 = 0

Dividing both sides by 2, we get:

x² + x - 20 = 0

This is a quadratic equation that can be factored:

(x + 5)(x - 4) = 0

The solutions are x = -5 and x = 4. Since the width of the rectangle cannot be negative, the only valid solution is:

x = 4

Substituting this value for "x" into the equation for "L", we get:

L = 2x + 2 = 2(4) + 2 = 10

Therefore, the length of the rectangle is 10 inches and the width is 4 inches.

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Answer:

2/15 litre

Step-by-step explanation:

There is 1 litre in total. Max used 1/5 litre.

1 - 1/5 = 4/5 litre

Then he used 5/6 of the remaining 4/5 litre. He would have 1/6 of 4/5 litre left.

1/6 * 4/5 = 2/15 litre

or

You could do 5/6 * 4/5 to find out how much syrup he used on Sunday and then subtract that from 4/5 litre. It will give you the same answer

The company now has 675 employees after adding 35% more employees to the initial 500 employees.

A figure or ratio that may be stated as a fraction of 100 is a percentage. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100.

To calculate the total number of employees after adding 35%, you can multiply the initial number of employees by 1.35:

Total number of employees = 500 x 1.35

Total number of employees = 675

Therefore, the company now has 675 employees after adding 35% more employees to the initial 500 employees.

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The result of the carrying out the operation, Add -4(row 1) to row 3, on the given matrix is:

[tex]\left[\begin{array}{ccc|c}1&2&1&-5\\0&4&-2&3\\0&-9&2&12\end{array}\right][/tex]

From the question, we are to perform the given row operation on the given matrix

The given matrix is:

[tex]\left[\begin{array}{ccc|c}1&2&1&-5\\0&4&-2&3\\4&-1&6&-8\end{array}\right][/tex]

The operation we are to carry out is:

Add -4(row 1) to row 3

This essentially means we should multiply the first row (row 1) by -4. Then, we will add each of the elements from the results to the corresponding elements on the third row

Multiplying by -4

-4 × [ 1     2    1   | -5

      -4   -8   -4  | 20

Now, we will add the result from the multiplication to the corresponding elements in row 3

    4    -1     6  |  -8

+  -4   -8   -4  | 20

------------------------

    0    -9    2  |  12

Hence,

The resulting matrix becomes

[tex]\left[\begin{array}{ccc|c}1&2&1&-5\\0&4&-2&3\\0&-9&2&12\end{array}\right][/tex]

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Given the function h(x) = -2√(x + 3) - 1, the statement that is true about h(x) include the following: B. the function is decreasing on the interval (-3, ∞).

For any given function, y = f(x), if the output value (range or y-value) is decreasing when the input value (domain or x-value) is increased, then, the function is generally referred to as a decreasing function.

For any given function, y = f(x), if the output value (range) is increasing when the input value (domain) is increased, then, the function is generally referred to as an increasing function.

By critically observing the graph of the given function, we can reasonably infer and logically deduce that it is decreasing over the interval (-3, ∞).

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Thus, new coordinates of A'B'C' after the reflection across the y-axis are-

A'(-5, -1), B'(-3,-4), C'(-3, 0).

The x-coordinate remains constant when a point is reflected across the x-axis, but the y-coordinate is assumed to be the additive inverse. Point (x, y) is reflected across the x-axis as (x, -y).

The y-coordinate stays the same when a point is reflected across the y-axis, but the x-coordinate is assumed to be the additive inverse. Point (x, y) is reflected across the y-axis as (-x, y).

Given coordinates of ABC.

A(5, -1), B(3,-4), C(3, 0)

Reflection across the y-axis : y coordinates remains same while x coordinate multiplied with -1.

(x,y) -- > (-x, y)

A'(-5, -1), B'(-3,-4), C'(-3, 0)

Thus, new coordinates of A'B'C' after the reflection across the y-axis are-

A'(-5, -1), B'(-3,-4), C'(-3, 0).

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part a:

Henry should take a route that bypasses the park on the way to school and on the way home from school in order to ride the least possible distance.

He should take Route C on his way to school and Route B on his back.

part b: no values were given to calculate the distance.

Distance is described as a numerical or occasionally qualitative measurement of how far apart objects or points are.

Since the park near Henry's house is closed before school, so he cannot ride through the park on his way to school, he should take Route C on his way to school and Route B on his back.

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If the yard is 2000 feet², both companies will charge the same price.

In mathematics, the word "amount" is a broad one that refers to the size or quantity of something, typically given as a numerical value.

It can be used in a number of circumstances where there is a concern with money, measurements, or the quantity of an item.

We must assume both organisations' expenses to be equal and then solve for the yard size to determine the point at which their costs are equal.

Let's assume that x is the yard size at which expenses are equal.

7.5x + 24.5 = 16x + 23.5

When we simplify this equation, we obtain:

0.5x = 1

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Answer: direction angle of c is pi/4.

Step-by-step explanation: We can find c by adding the corresponding components of a and b:

c = a + b = ⟨–7, 3⟩ + ⟨–2, –12⟩ = ⟨–9, –9⟩

To find the magnitude of c, we can use the formula:

|c| = sqrt(c1^2 + c2^2)

where c1 and c2 are the x- and y-components of c, respectively. In this case, we have:

|c| = sqrt((-9)^2 + (-9)^2) = sqrt(162) = 9sqrt(2)

To find the direction angle of c, we can use the formula:

theta = atan(c2 / c1)

where theta is the angle between the positive x-axis and the vector c. In this case, we have:

theta = atan((-9) / (-9)) = atan(1) = pi/4

So the direction angle of c is pi/4.

Therefore, the magnitude of c is 9sqrt(2) and the direction angle of c is pi/4.

Answer:

B

Step-by-step explanation:

The area of the rectangle is 27/70 square feet.

The area of a rectangle is given by the formula:

Area = base x height

Plugging in the given values, we get:

Area = (3/7) x (9/10)

Simplifying, we get:

Area = (27/70) square feet

Therefore, the area of the rectangle is 27/70 square feet.

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The slope-intercept form of the equation is y = 18x.

To find the function in slope-intercept form, we have to write the general form below as;

y = mx + c

m = slope

c = y-intercept

To find the slope, we need to take two points from the table given;

slope (m) = y2 - y1 / x2 - x1

m = 720 - 450 / 40- 25

m = 18

The slope of the line is 18

we can use this with any point to find the y - intercept

y = mx + c

720 = 18(40) + c

720 = 720 + c

c = 720 - 720

c = 0

The equation of the line in slope-intercept form is y = 18x

b. To determine how far he travelled with 55 gallons of gas

y = 18x

y = 18(55)

y = 990 miles

c. How many gallons will be required to cover 630 miles

y = 18x

630 = 18x

x = 630 / 18

x = 35 gallons

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The pair of the functions that are inverse is

The inverse of the function is solved by the operations as follows

f(x) = x - 4 let y = f(x)

y = x - 4

solving for x

x = y + 4

interchanging the variables

y = x + 4  (this is the inverse)

the inverse x + 4 is equal to g(x) hence they are inverse functions

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If the loan is a total of $8,465 at 5.9% compounded monthly. at the end of the 12-month repayment period, the student will owe a total of $9246.35 on the loan.

   First, calculate the monthly interest rate by dividing the annual interest rate by 12:

Monthly interest rate = 5.9% / 12 = 0.004917

   Next, calculate the number of months in the repayment period, including the grace period:

Total repayment period = 12 + 6 = 18 months

   Use the formula for calculating the future value of a loan to find the total amount owed at the end of the repayment period:

   FV = PV x (1 + r)^n

   where:

   PV = the present value of the loan (i.e. the amount borrowed)

   r = the monthly interest rate

   n = the number of months in the repayment period

   Plugging in the numbers, we get:

FV = $8,465 x (1 + 0.004917)^18

FV = $9246.35

Therefore, the student will owe a total of $9246.35 on the loan.

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The required unit rate is 0.333 products per minute.

What is a product's utilisation rate?

A measurement of how much of a product a user consumes in a certain amount of time; users can be classified as heavy, moderate, or light.

Given that a factory worker can make 15 products in 45 minutes.

We are to find the unit rate.

We will be using the unitary method to solve the given problem.

In 45 minutes, the number of products made by the worker = 15

Therefore, in 1 minute, the number of products made by the worker is given by

= Number of products / Amount of time taken

= 15/45

= 0.333

Thus, the required unit rate is 0.333 products per minute.

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We would need to study 678 families to determine the market preference for baby shampoo brands with a 96% confidence level and 4% margin of error, given that there are 12,000 families with babies in the sector of interest.

To calculate the sample size needed to determine the market preference for baby shampoo brands, we can use the formula:

n = [(Z^2 * p * q) / E^2]

where:

n = sample size

Z = Z-score for the desired level of confidence (in this case, 1.96 for a 96% confidence level)

p = proportion expected (in this case, 0.12 or 12%)

q = 1 - p

E = margin of error (in this case, 0.04 or 4%)

Substituting these values into the formula, we get:

n = [(1.96^2 * 0.12 * 0.88) / 0.04^2] = 677.16

Since we cannot have a fraction of a family, we round up the sample size to 678 families. Therefore, we would need to study 678 families to determine the market preference for baby shampoo brands with a 96% confidence level and 4% margin of error, given that there are 12,000 families with babies in the sector of interest.

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