Jake can run 4/5 of a mile in 1/10 of an hour. How many miles can he run per hour.

This equation represents a linear relationship between the total cost and the distance driven, with a slope of 0.25 and a y-intercept of 50.

If the initial fee is $50 and the variable cost is $0.25 per mile driven, the equation becomes:

y = 0.25x + 50

A linear relationship is a type of mathematical relationship that exists between two variables, where the change in one variable is directly proportional to the change in the other variable. In other words, a linear relationship can be expressed as a straight line when plotted on a graph.

Linear relationships are commonly used in many fields, including physics, economics, and engineering, to model real-world phenomena. They are often represented by linear equations, where the slope of the line represents the rate of change between the two variables. Linear relationships can also be used to make predictions or to estimate future values of one variable based on the value of the other variable.

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Complete Question:-

Derek is renting a van from Abby's Autos. He will be charged an initial fee to rent the van and an additional amount that depends on how far he drives it. 15 This situation can be modeled as a linear relationship.

Yes, all of the columns of an orthogonal matrix must be normal (or normalized). An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors. This means that the dot product of any pair of columns (or rows) is 0 if they are different, and 1 if they are the same.

Let's break this down:

1. Orthogonal unit vectors: The columns of an orthogonal matrix are orthogonal, meaning they are perpendicular to each other. They are also unit vectors, which means they have a magnitude (length) of 1.
2. Normalized: A vector is considered normalized if its magnitude is 1. Since the columns of an orthogonal matrix are unit vectors, they are also normalized.
3. Dot product: The dot product of two orthogonal unit vectors is 0 if they are different, and 1 if they are the same. This property is used to check if a matrix is orthogonal.

In summary, all of the columns of an orthogonal matrix must be normal (normalized), as they are orthogonal unit vectors.

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To determine how many ounces of mixed spices the chef put into each bowl, we need to divide the total amount of mixed spices by the number of bowls: the chef put [tex]1.5[/tex] ounces of mixed spices into each bowl.

The chef mixed 3 packages of spices, each containing 1 ounce of spices. Therefore, the total amount of spices mixed is:

3 packages   [tex]\times[/tex]  1 ounce/package = 3 ounces

The chef then put equal amounts of the mixed spices into 2 separate bowls. This means that the amount of mixed spices in each bowl is equal.

3 ounces / 2 bowls = 1.5 ounces per bowl

The chef has a total of 3 packages of spices, each containing 1 ounce of spices. So, the chef has a total o  [tex]3 \times 1 = 3[/tex] ounces of spices.

The chef puts an equal amount of mixed spices into each of the 2 separate bowls. Therefore, the chef puts 3/2 = 1.5 ounces of mixed spices into each bowl.

Therefore, the chef put  [tex]1.5[/tex] ounces of mixed spices into each bowl.

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To find the length of the arc of y=2/3x^3/2 from x=0 to x=3, we can use the formula, and Therefore, the length of the arc of y=2/3x^3/2 from x=0 to x=3 is approximately 8.01.

length = ∫[a,b] √[1 + (dy/dx)^2] dx
In this case, a = 0, b = 3, and dy/dx = (3/2)x^1/2. Plugging these values into the formula, we get:
length = ∫[0,3] √[1 + (3/2x^1/2)^2] dx
Simplifying the expression inside the square root, we get:
length = ∫[0,3] √[1 + 9/4x] dx
We can use the substitution u = 1 + 9/4x to simplify the integral:
u = 1 + 9/4x
du/dx = 9/4
dx = 4/9 du
When x = 0, u = 1, and when x = 3, u = 1 + 9/4(3) = 10.5. Substituting these values and the expression for dx into the integral, we get:
length = ∫[1,10.5] √u (4/9) du
Using the power rule of integration, we get:
length = (4/9) [2/3 u^(3/2)] [1,10.5]
Simplifying, we get:
length = (8/27) [10.5^(3/2) - 1^(3/2)]
length ≈ 8.01
Therefore, the length of the arc of y=2/3x^3/2 from x=0 to x=3 is approximately 8.01.

The length of the arc of y=(2/3)x^(3/2) from x=0 to x=3 can be found using the arc length formula:
Arc length = ∫(√(1 + (dy/dx)^2)) dx, with limits of integration from 0 to 3.
Step 1: Find the derivative of y with respect to x (dy/dx).
y = (2/3)x^(3/2)
dy/dx = (3/2)(2/3)x^(1/2) = x^(1/2)
Step 2: Square the derivative and add 1.
(dy/dx)^2 = (x^(1/2))^2 = x
1 + (dy/dx)^2 = 1 + x
Step 3: Find the square root of (1 + (dy/dx)^2).
√(1 + (dy/dx)^2) = √(1 + x)
Step 4: Integrate √(1 + (dy/dx)^2) with respect to x, from 0 to 3.
Arc length = ∫(√(1 + x)) dx, with limits of integration from 0 to 3.
Unfortunately, the integral of √(1 + x) does not have an elementary antiderivative. However, you can approximate the arc length using numerical integration methods, such as Simpson's Rule or the Trapezoidal Rule, or use a calculator or software capable of evaluating definite integrals numerically.

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Decomposition of 4/5 in two different ways is a) 1/5 + 1/5 + 1/5 + 1/5

                                                                            b) 1/5 + 3/5

Define decompose.

A fraction is a representation of a portion of a whole. Decomposing a fraction entails splitting it up into smaller pieces.

  The initial fraction must be obtained by adding together or combining all of the smaller or broken pieces.

The decomposition of a fraction can be done in two different ways:

Given:

4/5

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The second part of the trip will take 0.75 hours, or 45 minutes.

To solve this problem, we will need to use the terms "average speed," "distance," and "time."
First, let's find out how far the class travels during the 30-minute stop at the local wildlife refuge.

We can do this using the formula:
distance = average speed × time
In this case, the average speed is 50 miles per hour and the time is 0.5 hours (30 minutes converted to hours).
distance = 50 miles/hour × 0.5 hours = 25 miles
Now, let's find the remaining distance the class needs to travel to reach the natural history museum.

We can do this by subtracting the distance they traveled during the first part of the trip from the total distance:
remaining distance = total distance - distance traveled during the first part
remaining distance = 62.5 miles - 25 miles = 37.5 miles.
Finally, let's find out how long it takes for the class to travel the remaining distance.

We can use the same formula as before, but this time, we'll rearrange it to solve for time:
time = distance ÷ average speed
For the second part of the trip, the distance is 37.5 miles and the average speed is still 50 miles per hour:
time = 37.5 miles ÷ 50 miles/hour = 0.75 hours.

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Each tire made approximately 375 rotations on Stan's drive.

Distance covered by each tire = 3300 m

Distance is the amount of space between two points or objects. It can be measured in various units such as kilometers, miles, meters, feet, etc. In mathematics, distance is often calculated using the Pythagorean theorem or other distance formulas, depending on the context of the problem.

To find the number of rotations each tire made on Stan's drive, we need to first find the distance each tire covered and then divide it by the circumference of the tire.

The distance covered by each tire is equal to the distance Stan drove divided by the number of tires on his truck. Therefore,

distance covered by each tire = 3300 m / 4 = 825 m

The number of rotations each tire made is equal to the distance covered by the tire divided by its circumference. Therefore,

the number of rotations = distance covered by tire/circumference of the tire

For the given tire with a circumference of 2.2 m, we have

number of rotations = 825 m / 2.2 m ≈ 375

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Answer:

7

Step-by-step explanation:

55.89 can be estimated to be 56

7.68 can be estimated to be 8

56/8=7

Final Answer:   The small change in x and corresponding change in y  is             [tex]dy = (6x^2-2)dx[/tex]

                           dy = ((sec^211x)*11) dx

f(x) =  [tex]2x^3-2x[/tex]

Here we can consider y as function of x so y=f(x).

Derivative is nothing but rate of change i.e. small change in y with respect to x when y is function of x.  

[tex](dy/dx )= 6x^2-2[/tex]

[tex]dy = (6x^2-2)dx[/tex]

Second function here is f(x) = tan11x

Here also we will consider y as function of x

here we need to apply chain rule multiple of x is there.

so [tex](dy/dx) = (sec^211x)*11[/tex]

[tex]dy = ((sec^211x)*11) dx[/tex]

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Option 3) The number of red jelly beans expected in B and the number of black jelly beans expected in A are the same.

(b) Justification: Initially, Jar-A has 380 red jelly beans and Jar-B has 380 black jelly beans. When you scoop 20 red jelly beans from A and put them into B, Jar-B has 380 black jelly beans and 20 red jelly beans. After shaking, you scoop 20 jelly beans from B and put them into A. Since you took out 20 jelly beans from B, it will still have 380 jelly beans. The probability of picking a red jelly bean from B is now 20/380, so you can expect to return approximately

(20/380) × 20 = 20/19 ≈ 1 red jelly bean to A, leaving 19 red jelly beans in B.

Similarly, you can expect to have 20 - 1 = 19 black jelly beans in A,

as you scooped out 20 jelly beans from B. Therefore, the number of red jelly beans expected in B and the number of black jelly beans expected in A are the same.

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Aida and Marco need approximately 37 feet of painter's tape to cover the edges of both walls.

To find the length of the painter's tape needed to cover the edges of both walls, we need to find the perimeter of the trapezoid-shaped walls.

First, we need to find the lengths of the sides of the trapezoid-shaped walls. We can use the distance formula to find the lengths of the sides:

[tex]AB = \sqrt{[(6 - (-5))^2 + (3 - (-1))^2]}= \sqrt{(11^2 + 4^2) }= \sqrt{137BC = \sqrt{[(-5 - (-5))^2 + (-5 - 3)^2] }= \sqrt{(0^2 + (-8)^2)}= 8CD = \sqrt{[(6 - (-5))^2 + (-5 - (-1))^2] }= \sqrt{(11^2 + 16^2)} = \sqrt{317DA = \sqrt{[(-5 - 6)^2 + (-1 - (-5))^2] }= \sqrt{(11^2 + 16^2) }= \sqrt{317[/tex]

Next, we can find the perimeter of the trapezoid-shaped walls by adding up the lengths of the sides:

Perimeter = AB + BC + CD + DA

= √137 + 8 + √317 + √317

= √137 + 2√317 + 8

≈ 36.65 feet

Therefore, To find the length of the painter's tape needed to cover the edges of both walls, we need to find the perimeter of the trapezoid-shaped walls.

First, we need to find the lengths of the sides of the trapezoid-shaped walls. We can use the distance formula to find the lengths of the sides:

[tex]AB = \sqrt{[(6 - (-5))^2 + (3 - (-1))^2]}= \sqrt{(11^2 + 4^2) }= \sqrt{137BC = \sqrt{[(-5 - (-5))^2 + (-5 - 3)^2] }= \sqrt{(0^2 + (-8)^2)}= 8CD = \sqrt{[(6 - (-5))^2 + (-5 - (-1))^2] }= \sqrt{(11^2 + 16^2)} = \sqrt{317DA = \sqrt{[(-5 - 6)^2 + (-1 - (-5))^2] }= \sqrt{(11^2 + 16^2) }= \sqrt{317[/tex]

Next, we can find the perimeter of the trapezoid-shaped walls by adding up the lengths of the sides:

Perimeter = AB + BC + CD + DA

= √137 + 8 + √317 + √317

= √137 + 2√317 + 8

≈ 36.65 feet

Therefore, Aida and Marco need approximately 37 feet of painter's tape to cover the edges of both walls.

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1) Find c: To find the value of c, we need to make sure the PDF integrates to 1, as the total probability should always equal 1. So, we have: ∫[0,∞] ce^(-4x) dx = 1.

When we integrate, we get: -c/4 * (e^(-4x)) | [0,∞] = 1, Plugging in the limits, we get: (-c/4 * 0) - (-c/4 * 1) = 1, c/4 = 1, Thus, c = 4.(2). Find the CDF of X, F_X(x): To find the CDF, we integrate the PDF from the lower bound (0) to x: F_X(x) = ∫[0,x] 4e^(-4t) dt.

When we integrate, we get: F_X(x) = -e^(-4t) | [0,x], Plugging in the limits, we get: F_X(x) = (-e^(-4x) - (-e^0)) = 1 - e^(-4x)
3. Find P(2)

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The total cost of the camera with sales tax included is $811.47.

The total cost refers to the summation of all the cost involved from the manufacturing to sales, it majorly involves the fix cost and variable cost.

To find the total cost we need to derive a formula

total cost = listed price + ( listed price x sales tax rate)

now by placing the values given in the question we can calculate the the total cost

listed price = $761. 98

sales tax rate = 6. 5%

then,

total cost = 761.98 + (761.98 x 0.065)

total cost = 761.98 + 49.49

total cost = $811.47

The total cost of the camera with sales tax included is $811.47.

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29 students scored at least 70 but less than 90, which is evaluated by using the cumulative frequency.

To discover the number of understudies with scores more noteworthy than or rise to 70 but less than 90, we have to discover the cumulative frequency of scores between 70 and 90. 

From this table, ready to see that the total frequency of scores up to 70 is 63, and the total frequency of scores up to 90 is 92. So the number of understudies with scores over 70 and underneath 90 is:

Cumulative frequency of outcomes up to 90 - Cumulative frequency of outcomes up to 70

= 92 - 63

= 29

Therefore, 29 students scored at least 70 but less than 90.

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The centroid of the thin plate is located at (0, π/16).

To find the centroid of the thin plate lying in the region between the circle X² + Y²= 4 and inside the circle x² + y²=1 in the first quadrant, we need to use the formula for the centroid of a region in polar coordinates. First, we need to find the polar equation of the curves X²+ Y² = 4 and x² + y²=1.

The curve X² + Y² = 4 can be written as r = 2, and the curve x² + y²=1 can be written as r = 1.

Next, we need to find the limits of integration for r and theta. Since the thin plate lies in the first quadrant, the limits for theta are 0 and pi/2. For r, the limits are the equations of the two circles.

Therefore, the limits for r are 1 and 2.

Now we can use the formula for the centroid of a region in polar coordinates: x_c = (1/A) ∫∫ r² cos(theta) dr d(theta) y_c = (1/A) ∫∫ r^2 sin(theta) dr d(theta) where A is the area of the region.

To find the area of the region, we can use the formula:

A = ∫∫ r dr d(theta) Substituting the limits and simplifying, we get:

A = 3π/4

Now we can evaluate the integrals for x_c and y_c:

x_c = (1/3π/4) ∫0⁽π/²⁾∫1²r³ cos(theta) dr d(theta) = 0 y_c = (1/3π/4) ∫0^(π/2) ∫1² r³ sin(theta) dr d(theta) = π/16

Therefore, the centroid of the thin plate is located at (0, π/16).

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To find the population size after 4 hours, we can use the formula:

P(t) = P(0) * e^(rt)
where P(t) is the population size at time t, P(0) is the initial population size, r is the relative growth rate, and e is the mathematical constant approximately equal to 2.718.

Plugging in the given values, we get:

P(4) = 3.1 million * e^(0.4435 * 4)
P(4) = 3.1 million * e^(1.774)
P(4) = 3.1 million * 5.913
P(4) = 18.333 million cells
Therefore, the population size after 4 hours is 18.3 million cells (rounded to one decimal place).
 Based on the given information, the population of yeast cells grows with a constant relative growth rate of 0.4435 per hour, and the initial population is 3.1 million cells. To find the population size after 4 hours, we can use the exponential growth formula:
P(t) = P0 * e^(rt)

where P(t) is the population size after t hours, P0 is the initial population (3.1 million cells), r is the growth rate (0.4435 per hour), and e is the base of the natural logarithm (approximately 2.71828).
For this problem, t = 4 hours. Plugging the values into the formula, we get:
P(4) = 3.1 * e^(0.4435 * 4)
Now, calculate the exponent:
0.4435 * 4 = 1.774
Then, calculate e raised to this power:
e^1.774 ≈ 5.897
Next, multiply this by the initial population:
3.1 * 5.897 ≈ 18.2797

So, after rounding the answer to one decimal place, the population size after 4 hours is approximately 18.3 million cells.

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To solve this problem, we first need to find the derivative of g(x), which is g'(x) = 2x - 3. We are given that f(x) = g'(x), so f(x) = 2x - 3.

Next, we can use the definite integral to evaluate ∫ 1 3 f(x) dx. This is the area under the curve of f(x) between x = 1 and x = 3.

To find this area, we can use the formula for the definite integral:

∫ 1 3 f(x) dx = [F(x)] from 1 to 3

where F(x) is the antiderivative of f(x). Since f(x) = 2x - 3, we can integrate this to get F(x) = x^2 - 3x.

Evaluating the integral at x = 3 and x = 1, we get:

[F(x)] from 1 to 3 = F(3) - F(1)

= (3^2 - 3(3)) - (1^2 - 3(1))

= 0

Therefore, the answer is (B) -2.
To solve the problem, first we need to find the derivative of g(x) which is f(x):

g(x) = x^2 - 3x + 4
g'(x) = f(x) = 2x - 3

Now we need to find the definite integral of f(x) from 1 to 3:

∫[1, 3] f(x) dx = ∫[1, 3] (2x - 3) dx

To find the integral, use the power rule for integration:

∫(2x - 3) dx = x^2 - 3x + C

Now, apply the limits of integration:

(x^2 - 3x) | [1, 3] = (3^2 - 3*3) - (1^2 - 3*1) = (9 - 9) - (1 - 3) = 0 + 2

So, the answer is (C) 2.

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The length of the curve defined by x = y * sqrt(5), where 4 ≤ y ≤ 6, is 2 sqrt(6/5).

To find the length of the curve defined by the equation x = y * sqrt(5), where 4 ≤ y ≤ 6, we can use the arc length formula:

L = ∫(a to b) [tex]sqrt[1 + (dy/dx)^2] dx[/tex]

First, we need to find [tex]dy/dx[/tex]. We can start by implicitly differentiating x = y * sqrt(5) with respect to y:

x = y * sqrt(5)

=> [tex]1 = sqrt(5) * dy/dx[/tex]

Solving for [tex]dy/dx[/tex], we get:

[tex]dy/dx = 1/sqrt(5)[/tex]

Now we can substitute this into the arc length formula and integrate from y = 4 to y = 6:

L = ∫(4 to 6) [tex]sqrt[1 + (dy/dx)^2][/tex] dx

= ∫(4 to 6)  [tex]sqrt[1 + (1/sqrt(5))^2][/tex] dx

= ∫(4 to 6) sqrt[1 + 1/5] dx

= ∫(4 to 6) sqrt[6/5] dx

= sqrt(6/5) * ∫(4 to 6) dx

= sqrt(6/5) * (6 - 4)

= sqrt(6/5) * 2

= 2 sqrt(6/5)

Therefore, the length of the curve defined by x = y * sqrt(5), where 4 ≤ y ≤ 6, is 2 sqrt(6/5).

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a. 19 in eight-bit signed two's-complement representation is 00010011.

b. -19 in eight-bit signed two's-complement representation is 11101101.

c. 75 in eight-bit signed two's-complement representation is 01001011.

d. -87 in eight-bit signed two's-complement representation is 10101001.

e. -95 in eight-bit signed two's-complement representation is 10100001.

f. 99 in eight-bit signed two's-complement representation is 01100011.

Two's complement is a mathematical operation used in digital electronics and computer arithmetic to represent signed numbers.

a. 19 in binary is 00010011. Since 19 is a positive number, its eight-bit signed two's-complement representation is simply 00010011.

b. To find the eight-bit signed two's-complement representation of -19, we first need to convert 19 to binary (00010011) and then flip all the bits to get 11101100. This is the one's complement of 00010011. Next, we add one to the one's complement to get the two's complement, which is 11101101. Therefore, the eight-bit signed two's-complement representation of -19 is 11101101.

c. 75 in binary is 01001011. Since 75 is a positive number, its eight-bit signed two's-complement representation is simply 01001011.

d. To find the eight-bit signed two's-complement representation of -87, we first need to convert 87 to binary (01010111) and then flip all the bits to get 10101000. This is the one's complement of 01010111. Next, we add one to the one's complement to get the two's complement, which is 10101001. Therefore, the eight-bit signed two's-complement representation of -87 is 10101001.

e. To find the eight-bit signed two's-complement representation of -95, we first need to convert 95 to binary (01011111) and then flip all the bits to get 10100000. This is the one's complement of 01011111. Next, we add one to the one's complement to get the two's complement, which is 10100001. Therefore, the eight-bit signed two's-complement representation of -95 is 10100001.

f. 99 in binary is 01100011. Since 99 is a positive number, its eight-bit signed two's-complement representation is simply 01100011.

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The answer is 3) 24.

An arithmetic progression or arithmetic sequence (AP) is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that arithmetic progression. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2.

To find the number of different ways to rearrange the letters in the word "math" to form a four-letter code word, we need to use the permutation formula, which is n! / (n-r)!, where n is the total number of items and r is the number of items being selected. In this case, n is 4 (the number of letters in the word "math") and r is also 4 (since we are forming a four-letter code word).

So the number of different ways to rearrange the letters in "math" is 4! / (4-4)! = 4! / 0! = 24. Therefore, the answer is 3) 24.

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Using constant of proportionality, when x = 3 and y = 4, the value of z is 8.

What is a proportionality?

Proportionality is a mathematical relationship between two variables, stating that they change in a constant ratio. When two variables are proportional, as one variable increases or decreases, the other variable changes by a corresponding factor.

In mathematical terms, we say that two variables, x and y, are proportional if their ratio is constant:

x/y = k

Now,

If x varies directly as y and inversely as z, we can write the equation:

x = k * (y/z)

where k is a constant of proportionality.

To find the value of k, we can use the initial conditions given:

x = 10,y = 5, z = 3

10 = k * (5/3)

k = 6

Now, we can use the equation to find the value of z when x = 3 and y = 4:

3 = 6 * (4/z)

z = 8

Therefore, when x = 3 and y = 4, the value of z is 8.

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Answer:

1. 18

2. 16

3. 45

4. 72

5. 32

6. 42

7. 20

8. 9

9. 78

10. 4

Answer:

d) y = (-3/4)(x - 3)^2 - 7

Step-by-step explanation:

The vertex form of a quadratic function is:

y = a(x - h)^2 + k

where (h, k) is the vertex of the parabola.

We are given that the vertex is (3, -7), so we can substitute these values into the equation:

y = a(x - 3)^2 - 7

Now we need to find the value of "a". We can use the fact that the function passes through the point (1, -10). Substituting these values into the equation gives us:

-10 = a(1 - 3)^2 - 7

Simplifying, we get:

-10 = 4a - 7

-3 = 4a

a = -3/4

Substituting this value of "a" into the equation, we get:

y = (-3/4)(x - 3)^2 - 7

Therefore, the quadratic function in vertex form that can be represented by the graph that has a vertex at (3, -7) and passes through the point (1, -10) is:

y = (-3/4)(x - 3)^2 - 7

The amount of money paid for the rope in dollars per inch is 0.025 $/in.

Based on the information provided above, we can logically deduce that the store bought 300 feet of nylon rope.

Conversion:

1 feet = 12 inches.

300 feet × 12 inches/feet = 3600 inches.

Therefore, we now know that this store bought 3600 inches of nylon rope.

Furthermore, this store sold 40 inches of nylon rope for $1.60. Thus, we can calculate the selling unit price of nylon rope in dollars per inch as follows;

Selling unit price = $1.60/(40 in.)

Selling unit price = $0.04/in.

At a price of $0.04 per inch, we have:

Selling Price =3600 inches × $0.04/in.

Selling Price = $144

Therefore, since the profit was $54 for selling all of the nylon rope;

Cost price = Selling price - profit

Cost price = $144 - $54

Cost price = $90

The cost in dollars per inch is given by;

Cost in dollars per inch = ($90)/(3600 in.)

Cost in dollars per inch = 0.025 $/in.

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Answer:

Answer B is correct

Step-by-step explanation:

This is a trapezium.

The formula to find the area of a trapezium is,

1/2 ( Sum of the parallel sides ) × height

Let us find it now.

[tex]\sf \frac{1}{2}*(38+18)*16 \\\\\sf \frac{1}{2}*56*16 \\\\\sf \frac{1}{2}*896\\\\448cm^2[/tex]

The Laplace transform of y''(t) + 4y(t) = sin(nt) with initial values y(0) = 0 and y'(0) = 0 is Y(s) = sin(nt)/(s^2 + 4).

Using the Laplace transform, we can write:

s^2 Y(s) - s*y(0) - y'(0) + 4Y(s) = 1/(s^2 + n^2)

Substituting y(0) = 0 and y'(0) = 0, we get:

s^2 Y(s) + 4Y(s) = 1/(s^2 + n^2)

Factoring out Y(s), we have:

Y(s) = sin(nt)/(s^2 + 4)

To find y(t), we need to take the inverse Laplace transform of Y(s). Using a table of Laplace transforms, we know that the inverse Laplace transform of sin(nt)/(s^2 + 4) is:

y(t) = u(t)*sin(2t)/2

where u(t) is the unit step function.

Therefore, the solution to the differential equation y''(t) + 4y(t) = sin(nt) with initial values y(0) = 0 and y'(0) = 0 is y(t) = u(t)*sin(2t)/2.

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The length of material that Mrs. Rodriguez have left to use on the smaller dress is calculated to be  521 [tex]\frac{2}{3}[/tex] yards of material left to use on the smaller dress.

To determine how much material Mrs. Rodriguez will have left to use on the smaller dress, we need to subtract the amount of material used for the larger dress from the total amount of material available.

First, we need to convert the mixed number of the larger dress material requirement to an improper fraction:

3 [tex]\frac{23}{3}[/tex] = (3 x 3 + 23)/3 = 32/3

So, the larger dress requires 32/3 yards of material.

Next, we can subtract this amount from the total amount of material available:

613 [tex]\frac{1}{3}[/tex] - 32/3

To subtract these two values, we need to have a common denominator:

613 [tex]\frac{3}{9}[/tex] - 32/3

Now we can subtract the fractions:

= 613 [tex]\frac{23}{3}[/tex]- 96/9

= 617 [tex]\frac{2}{3}[/tex] - 96/9

= 521 [tex]\frac{2}{3}[/tex]

Therefore, Mrs. Rodriguez will have 521 [tex]\frac{2}{3}[/tex] yards of material left to use on the smaller dress.

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Using Chebyshev's theorem for standard deviation, at least 96% of the data lies within five standard deviations of the mean.

Chebyshev's theorem states that for any data set, regardless of its distribution, at least 1 - (1/k^2) of the data falls within k standard deviations from the mean.
Step 1: Identify the number of standard deviations (k) from the mean.
In this case, k = 5.
Step 2: Apply Chebyshev's theorem.
Chebyshev's theorem states that the proportion of data within k standard deviations of the mean is at least (1 - 1/k^2).
Step 3: Plug in the value of k into the formula.
Percentage of data within 5 standard deviations = (1 - 1/5^2) = (1 - 1/25)
Step 4: Calculate the percentage.
Percentage of data within 5 standard deviations = (1 - 1/25) = 24/25 = 0.96
Step 5: Convert the proportion to a percentage.
0.96 * 100 = 96%
So, using Chebyshev's theorem for standard deviation, at least 96% of the data lies within five standard deviations of the mean.

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The inverse of 3 × f(x) by dividing the input of the original function f(x) by 3 and then applying the inverse function f-1(x) to the result.

On the other hand, inverse variation describes a relationship between two variables in which one variable rises while the other falls in proportion. In other words, if two variables x and y have an inversely proportional relationship, we can express this relationship as y = k/x, where k is a proportionality constant. As a result, if we double the value of x, the value of y will be halved, and if we reduce the value of x by half, the value of y will increase by two. Speed and time, as well as pressure and volume, are two examples of inverse variation.

To find the inverse of the function f(x), we need to first solve for x in terms of y, and then replace y with f-1(x) to obtain the inverse function.

Let y = 3 × f(x)

Dividing both sides by 3, we get:

y/3 = f(x)

Replacing f(x) with y/3, we get:

x = f-1(y/3)

Therefore, the inverse of 3 × f(x) is:

(f-1)(y) = x/3 = f-1(y/3)

In other words, we can obtain the inverse of 3 * f(x) by dividing the input of the original function f(x) by 3 and then applying the inverse function f-1(x) to the result.

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The answer doesn't exist for lim x → 0 tan−1(ln(x)).

To find the limit of the function as x approaches 0, lim(x→0) arctan(ln(x)), we need to analyze the behavior of the function near x = 0.

1. Consider the domain of the function. The natural logarithm ln(x) is only defined for x > 0, so we cannot evaluate the function at x = 0. However, we can still analyze the limit as x approaches 0 from the right.

2. As x approaches 0 from the right (x → 0+), the natural logarithm ln(x) approaches negative infinity (ln(x) → -∞).

3. The arctan function is continuous and has horizontal asymptotes at y = -π/2 and y = π/2. When its argument approaches negative infinity (ln(x) → -∞), the arctan function approaches its horizontal asymptote at y = -π/2.

Therefore, the limit of the function as x approaches 0 from the right is -π/2: lim(x→0+) arctan(ln(x)) = -π/2.

Since the function is not defined for x ≤ 0, the limit as x approaches 0 does not exist. Your answer is DNE (does not exist).

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