On the same coordinate plane, mark all points (x,y) such that (A) y=x-2, (B) y=-x-2, (C) y=|x|-2

The average number of vehicles registered in each borough of NYC is 4.2 x 10^5.

To find the average number of vehicles registered in each borough of NYC, we need to divide the total number of registered vehicles by the number of boroughs. Therefore, the average number of vehicles registered in each borough can be calculated as:

Average number of vehicles = Total number of vehicles registered / Number of boroughs

= 2.1 x 10^6 / 5

= 4.2 x 10^5

Therefore, the average number of vehicles registered in each borough of NYC is 4.2 x 10^5.

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The probability of choosing exactly 13 students who dressed up as the 40s out of the 16 selected students is approx. 0.4334.

We can model this situation as a hypergeometric distribution, where we have a population of 19 students, 15 of whom dressed up as the 40s.

We want to choose a sample of 16 students and find the probability that exactly 13 of them dressed up as the '40s.

The probability of choosing exactly 13 students who dressed up as the 40s can be calculated:

(number of ways to choose 13 students who dressed up as the 40s) * (number of ways to choose 3 students who dressed up as other decades) / (total number of ways to choose 16 students)

The number of ways to choose 13 students who dressed up as the '40s is the number of combinations of 15 choose 13:

(15 choose 13) = 105

The number of ways to choose 3 students who dressed up as other decades is the number of combinations of 4 choose 3, which is:

(4 choose 3) = 4

The total number of ways to choose 16 students from 19 is the number of combinations of 19 choose 16, which is:

(19 choose 16) = 969

105 * 4 / 969 = 0.4334

Therefore, the probability of choosing exactly 13 students who dressed up as the 40s = 0.4334 (rounded to four decimal places)

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Answer:  10/12

Step-by-step explanation:

since they give you adjacent to angle m and hypotenuse use

cos x = opp/hyp

cos M = 10/12

The evaluated probability that there have been 2 occurrences in ten minutes  is 0.0842, under the condition that the mean number of occurrences in ten minutes is 5.

Here we have to apply  the Poisson distribution formula. The formula is
[tex]P(X = k) = (e^{-g} * g^k) / k!,[/tex]
Here
X = number of occurrences,
k = number of occurrences we want to find the probability for,
e = Number of Euler's
g = mean number of occurrences in ten minutes.

For the given case, g = 5 since
Therefore,
P(X = 2) = (e⁻⁵ × 5²) / 2!
≈ 0.0842.

Hence,  after careful consideration the  evaluated probability that there are 2 occurrences in ten minutes is  0.0842.

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The p-value is less than the significance level so reject the null hypothesis and concluded percentage of the students volunteer their time is different from receiving financial aid students.

Percentage of college students who volunteer their time = 20%

Perform a hypothesis test.

Null hypothesis H₀: p = 0.20,

where p is the proportion of college students who volunteer their time.

The alternative hypothesis is Hₐ: p ≠ 0.20.

Indicating that the proportion of college students who volunteer their time is different for students receiving financial aid.

57 out of 381 randomly selected students who receive financial aid volunteered their time.

Test the hypothesis,

Calculate the sample proportion of volunteers among the students receiving financial aid,

p₁ = 57 / 381

    = 0.149

Using Test statistic,

which follows a normal distribution under the null hypothesis .

Mean = 0  

Standard deviation σ = √(p(1-p)/n),

where p = 0.20 is the proportion under the null hypothesis

n = 381 is the sample size.

z

= (p₁ - p) /√(p×(1-p)/n)

= (0.149 - 0.20) / √(0.20(1-0.20)/381)

= -2.55

Using attached table of p-value from z-score.

Calculated test statistic of -2.55 corresponds to a p-value of 0.0054,

which is less than the significance level α = 0.01.

Reject the null hypothesis .

Therefore, we conclude that there is evidence to suggest that the percentage of college students who volunteer their time is different for students receiving financial aid.

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The above question is incomplete, the complete question is:

20% of all college students volunteer their time. is the percentage of college students who are volunteers different for students receiving financial aid? of the 381 randomly selected students who receive financial aid, 57 of them volunteered their time. what can be concluded at the α = 0.01 level of significance?

The sample space for Ella's compound event where she rolls a die and then flips a coin can be represented in the table below:

Die: 1 2 3 4 5 6
Coin: H-1 H-2 H-3 H-5 H-6 T-1 T-3 T-4 T-5

The size of the sample space is the total number of possible outcomes, which in this case is the number of rows in the table. We can see that there are 9 rows in the table, so the size of the sample space is 9.

To understand the sample space, we can imagine that each row in the table represents a possible outcome of the compound event. For example, the first row represents the outcome where Ella rolls a 1 on the die and gets heads on the coin. The second row represents the outcome where Ella rolls a 2 on the die and also gets heads on the coin, and so on.

Understanding the sample space is important in probability theory because it allows us to calculate the probability of specific events occurring. By knowing the size of the sample space and the number of favorable outcomes, we can determine the probability of an event happening.

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If Jesse can mow 3 yards in 8 hours. Jackson can mow twice as many yards per hour the constant of proportionality between the number of yards Jackson can mow and the number of hours is 3/4.

To find the constant of proportionality between the number of yards Jackson can mow and the number of hours, we can use the formula:

k = y/x

where k is the constant of proportionality, y is the number of yards, and x is the number of hours.

We know that Jesse can mow 3 yards in 8 hours, which means his rate of mowing is: 3 yards/8 hours = 3/8 yards per hour

We also know that Jackson can mow twice as many yards per hour as Jesse, which means his rate of mowing is:

2 * (3/8) yards per hour = 3/4 yards per hour

Now we can use the formula to find the constant of proportionality for Jackson:

k = y/x = (3/4) yards per hour / 1 hour = 3/4

Therefore, the constant of proportionality between the number of yards Jackson can mow and the number of hours is 3/4.

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Answer:

IT depends on how many kids there are per group.

Step-by-step explanation:

At the point P(2,2), the unit vector for the direction of steepest ascent is  (-i + j)/√2, and the unit vector for the direction of steepest descent is (i - j)/√2. A vector that points in the direction of no change in the function at P is (2e^(-1/3)/49) i + (2e^(-1/3)/49) j + (2/7) k.

To find the unit vectors that give the direction of steepest ascent and steepest descent at P, we need to find the gradient of F at P and normalize it to obtain a unit vector.

First, we find the partial derivatives of F with respect to x and y

Fx = -2x e^(-x^2/(16-y^2))/((16-y^2)^2)

Fy = 2y e^(-x^2/(16-y^2))/((16-y^2)^2)

Plugging in the coordinates of P, we get

Fx(2,2) = -2e^(-1/3)/49

Fy(2,2) = 2e^(-1/3)/49

Therefore, the gradient of F at P is

∇F(2,2) = (-2e^(-1/3)/49) i + (2e^(-1/3)/49) j

To obtain the unit vector in the direction of steepest ascent, we normalize the gradient

u = (∇F(2,2))/||∇F(2,2)|| = (-i + j)/√2

To obtain the unit vector in the direction of steepest descent, we take the negative of u

v = -u = (i - j)/√2

To find a vector that points in a direction of no change in the function at P, we need to find a vector orthogonal to the gradient of F at P. One way to do this is to take the cross product of the gradient with the vector k in the z-direction

w = ∇F(2,2) x k = (2e^(-1/3)/49) i + (2e^(-1/3)/49) j + (2/7) k

Therefore, the vector that points in a direction of no change in the function at P is

(2e^(-1/3)/49) i + (2e^(-1/3)/49) j + (2/7) k

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Therefore, the distance x that a plant needing full sun can be placed from a fence that is 10.5 feet high is approximately 28.7 feet.

In order to find the distance x that a plant needing full sun can be placed from a fence that is 10.5 feet high, we will use the angle of elevation and the tangent function.

1. Given the lowest angle of elevation of the sun in winter is 20 degrees, we will use this angle in our calculations.

2. Set up a right triangle with the fence as the vertical side (opposite side), the distance x as the horizontal side (adjacent side), and the angle of elevation (20 degrees) at the point where the fence meets the ground.

3. Use the tangent function to find the distance x:

tan(angle) = opposite side / adjacent side

4. Plug in the values we have:

tan(20) = 10.5 / x

5. Solve for x:

x = 10.5 / tan(20)

6. Calculate the value of x:

x ≈ 28.7 feet

Therefore, the distance x is approximately 28.7 feet.

Note: The question is incomplete. The complete question probably is: In one area, the lowest angle of elevation of the sun in winter is 20 degrees. Find the distance x that a plant needing full sun can be placed from a fence that is 10.5 feet high. Round your answer to the tenths place when necessary.

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C(12) represents the total cost (in dollars) of renting the industrial cleaning unit for 12 hours.

The problem gives us a function C(t) = 60 + 24t, where t represents the number of hours that an industrial cleaning unit is rented for. The function tells us that the total cost (in dollars) of renting the unit is equal to $60 plus $24 per hour.

Now, we are asked to find what C(12) represents. To do so, we substitute t = 12 into the function, which gives us:

C(12) = 60 + 24(12)

We can simplify this expression by multiplying 24 by 12, which gives us:

C(12) = 60 + 288

Adding 60 and 288 together, we get:

C(12) = 348

So, C(12) represents the total cost (in dollars) of renting the industrial cleaning unit for 12 hours. Therefore, the correct answer to the question is: The cost of renting the unit for 12 hours.

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1) To find the differentials of z = x^2 - xy^2 + 4y^5, we can use the total differential formula:

dz = (∂z/∂x)dx + (∂z/∂y)dy

Taking the partial derivatives of z with respect to x and y:

∂z/∂x = 2x - y^2

∂z/∂y = -2xy + 20y^4

Substituting these into the total differential formula:

dz = (2x - y^2)dx + (-2xy + 20y^4)dy

2) To find the differentials of f(x,y) = (3x-y)/(x+2y), we can again use the total differential formula:

df = (∂f/∂x)dx + (∂f/∂y)dy

Taking the partial derivatives of f with respect to x and y:

∂f/∂x = (y-3)/(x+2y)^2

∂f/∂y = (3x-2y)/(x+2y)^2

Substituting these into the total differential formula:

df = [(y-3)/(x+2y)^2]dx + [(3x-2y)/(x+2y)^2]dy

3) To find the differentials of f(x,y) = xe^x3y, we can once again use the total differential formula:

df = (∂f/∂x)dx + (∂f/∂y)dy

Taking the partial derivatives of f with respect to x and y:

∂f/∂x = e^(x3y) + 3xye^(x3y)

∂f/∂y = 3x^2e^(x3y)

Substituting these into the total differential formula:

df = (e^(x3y) + 3xye^(x3y))dx + (3x^2e^(x3y))dy

Here are the results:

1) For z = x^2 - xy^2 + 4y^5, the partial derivatives are:
∂z/∂x = 2x - y^2
∂z/∂y = -2xy + 20y^4

2) For f(x,y) = (3x-y)/(x+2y), the partial derivatives are:
∂f/∂x = (3(x+2y) - 3(3x-y))/(x+2y)^2
∂f/∂y = (-1(x+2y) + (x+2y))/(x+2y)^2

3) For f(x,y) = xe^(x^3y), the partial derivatives are:
∂f/∂x = e^(x^3y) * (1 + 3x^2y)
∂f/∂y = xe^(x^3y) * x^3

These partial derivatives represent the differentials for each respective function.

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Answer:

the third one

Step-by-step explanation:

Answer:

Option 3 is the correct answer

Step-by-step explanation:

The surface area of a prism is the area of the full net.

The area of the full net is the sum of the areas of each part

For the given net, there are three rectangles, and two triangles.

The area for rectangles and triangles are given by the following formulas:

[tex]A_{rectangle}=base*height[/tex]

[tex]A_{triangle}=\frac{1}{2}*base*height[/tex]

It is important to recognize that due to the fact that the 3-D shape is a prism, the two triangles are congruent, and have exactly the same dimensions and area.

Looking at the options:

Option 1 has three products added together.  This would be the base time height of each of the three rectangles.  It does not include the area for either of the triangles.

Option 2 does have an extra term in front with 3 numbers multiplied together.  It most closely resembles 2 times the product of the base and height of the triangle, but recall the area for a triangle is one-half of the base times height (this may make more sense when looking at option 3).  This over-calculates the area of the triangle, and then doubles that over-calculated area (to match the second triangle)

Option 3 has an extra term in front with the number 2 times a parenthesis with 3 terms.  These three terms represent the "one-half" from the formula for the area of a triangle, and the base and height of the triangle.  The 2 in front of the parentheses represents that there are two of those triangles, both with that area.  This correctly calculates the area of the net, and thus, the surface area of the triangular prism.

Option 4 has an extra term in front, similar to option 3 which calculates the area of one triangle correctly, but fails to account for the area of the second triangle.

Option 3 is the correct answer.

Answer:

y=4x+4

Step-by-step explanation:

The slope formula is:
[tex]y=mx+b[/tex]

with m being the slope and b being the y-intercept

Given: slope=4, y-intercept=4

We can substitute the slope and the y-intercept into the question:

y=4x+4

Hope this helps! :)

The equation of the line in slope-intercept form of a line with slope 4 and y-intercept 4 is [tex]\text{y} = 4\text{x} + 4[/tex].

The ratio that y increase as x increases is the slope of a line. The slope of a line reflects how steep it is, but how much y increases as x increases. Anywhere on the line, the slope stays unchanged (the same).

[tex]\text{m}=\dfrac{\text{y}_2-\text{y}_1}{\text{x}_2-\text{x}_1}[/tex]

[tex]\text{m}=\dfrac{(\text{y}\bar{\text{a}}-\text{y}\bar{\text{a}})}{(\text{x}\bar{\text{a}}-\text{x}\bar{\text{a}})}[/tex]

It is given that:

A line with slope 4 and y-intercept 4.

The linear equation in one variable can be made:

As we know,

The standard equation of the line is:

[tex]\text{y} = \text{mx} + \text{c}[/tex]

Here m is the slope and c is the y-intercept.

[tex]\text{m} = 4[/tex]

[tex]\text{c} = 4[/tex]

[tex]\boxed{\bold{y = 4x + 4}}[/tex]

Thus, the equation of the line in slope-intercept form of a line with slope 4 and y-intercept 4 is [tex]\text{y} = 4\text{x} + 4[/tex].

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The length of side a is 50 if the angle ∠bac is 50° and the length of side b is 20 and side c is 35 using cosine law.

Length of side b = 20

Length of side c = 35

Angle ∠bac =  50°

To calculate the length of the side a, we need to use the cosine law. The formula is:

[tex]a^2 = b^2 + c^2 - 2bc cos(A)[/tex]

Substituting the given values in the formula, we get:

[tex]a^2 = 20^2 + 35^2 - 2(20)(35)cos(50°)[/tex]

[tex]a^{2}[/tex] = 400 + 1225 + (1400)*(0.642)

[tex]a^{2}[/tex] = 1625 + 898.8

a = [tex]\sqrt{2523.8}[/tex]

a = 50

Therefore we can conclude that the length of side a is 50 using cosine law.

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The depreciation for the first year of the truck's useful life is $7,467 by the straight-line method and $2,720 by the units-of-output method.

Depreciation per year = (Cost - Salvage value) / Useful life

Depreciation per year = (40,000 - 5,000) / 5 = $7,000

Depreciation for the first year = (16,000 / 15,000) x $7,000 = $7,467

Depreciation per mile = (Cost - Salvage value) / Total miles expected to be driven

Depreciation per mile = (40,000 - 5,000) / (5 x 15,000) = $0.17/mile

Depreciation for the first year = 16,000 x $0.17 = $2,720

Therefore, the depreciation for the first year of the truck's useful life is $7,467 by the straight-line method and $2,720 by the units-of-output method.

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given:  function f(x) = 2sin(x) + 3x + 3 ,Xo=1.5

1. Compute the derivative of the function, f'(x).
2. Use the iterative formula: Xₖ₊₁ = Xₖ - f(Xₖ) / f'(Xₖ)
3. Repeat the process 10 times.

First, let's find the derivative of f(x):
f'(x) = 2cos(x) + 3

Now, use the iterative formula to compute the iterations:

X₁ = X₀ - f(X₀) / f'(X₀)
X₂ = X₁ - f(X₁) / f'(X₁)
...
X₁₀ = X₉ - f(X₉) / f'(X₉)

Remember to not round any values until the final answer, and then round to six decimal places. Since I cannot actually compute the iterations, I encourage you to use a calculator or program to find the values for each Xₖ using the provided formula.

The probability of rolling a sum of 6 when two standard six-sided dice are rolled is 5/36, or approximately 0.139.

The probability of an event occurring is defined by probability. There are many instances in real life where we may need to make predictions about how something will turn out.

There are 36 possible outcomes when two standard six-sided dice are rolled. Each die has 6 possible outcomes, so the total number of outcomes is 6 x 6 = 36.

To find the probability of rolling a sum of 6, we need to count the number of ways we can get a sum of 6. There are five possible ways to get a sum of 6:

- Roll a 1 on the first die and a 5 on the second die

- Roll a 2 on the first die and a 4 on the second die

- Roll a 3 on the first die and a 3 on the second die

- Roll a 4 on the first die and a 2 on the second die

- Roll a 5 on the first die and a 1 on the second die

So, the probability of rolling a sum of 6 is 5/36.

Therefore, the probability of rolling a sum of 6 when two standard six-sided dice are rolled is 5/36, or approximately 0.139.

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The distance the boat travels to get to the beach is approximately 5.0 miles.

To see why, we can draw a right triangle on the coordinate plane, with one leg along the x-axis (going 3.5 miles east) and the other leg along the y-axis (going 4 miles south). The hypotenuse of this triangle is the straight distance from the marina to the beach, which is the distance the boat travels.

Using the Pythagorean theorem, we can find the length of the hypotenuse:

c^2 = a^2 + b^2

c^2 = (3.5)^2 + (4)^2

c^2 = 12.25 + 16

c^2 = 28.25

c ≈ 5.0

Therefore, the distance the boat travels to get to the beach is approximately 5.0 miles.

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The square root of units is sqrt (80), and the area of the kite to the nearest unit is 74 square units.

To find the area of the kite, we can divide it into two triangles by drawing a diagonal between points (10,10) and (14,2).
First, we need to find the length of this diagonal. We can use the distance formula:

d = sqrt((14-10)^2 + (2-10)^2)
d = sqrt(16 + 64)
d = sqrt(80)

So the length of the diagonal is square root(80) units.

Next, we can find the area of each triangle:

Triangle 1:
Base = 10 units
Height = 10 units
Area = 1/2 * base * height = 1/2 * 10 * 10 = 50 square units

Triangle 2:
Base = 6 units (the difference between the x-coordinate plane of (10,10) and (14,2))
Height = 8 units (the difference between the y-coordinate plane of (10,10) and (16,8))
Area = 1/2 * base * height = 1/2 * 6 * 8 = 24 square units

So the total area of the kite is 50 + 24 = 74 square units.

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Answer:    

On a coordinate plane, kite G D E F has points (0, 0), (10, 10), (16, 8), (14, 2).

Complete the steps to find the area of the kite.

What is GE?

Square root of

units

What is DF?

Square root of

units

What is the area of the kite to the nearest unit?

square units

The integral ∫8(1-tan²(x)/sec² dx Note Use an upper-case "C" for the constant of integration is ∫8(1-tan²(x)/sec²(x)) dx = 8 tan(x) + C where C is the constant of integration.

To evaluate the integral ∫8(1-tan²(x)/sec²(x)) dx, we need to use trigonometric identities to simplify the integrand.

First, we use the identity tan²(x) + 1 = sec²(x) to rewrite the integrand as follows:

8(1 - tan²(x)/sec²(x)) = 8(sec²(x)/sec²(x) - tan²(x)/sec²(x))

Simplifying this expression by canceling out the common factor of sec²(x), we get:

8(sec²(x) - tan²(x))/sec²(x)

Next, we use the identity sec²(x) = 1 + tan²(x) to simplify the expression further:

8(sec²(x) - tan²(x))/sec²(x) = 8((1 + tan²(x)) - tan²(x))/sec²(x)

Simplifying the expression inside the parentheses, we obtain:

8/ sec²(x)

Therefore, the integral simplifies to:

∫8(1-tan²(x)/sec²(x)) dx = ∫8/ sec²(x) dx

We can now use the substitution u = cos(x) and du/dx = -sin(x) dx to transform the integral into a simpler form:

∫8/ sec²(x) dx = ∫8/cos²(x) dx = 8∫cos(x)² dx

Using the power-reducing formula cos²(x) = (1 + cos(2x))/2, we get:

8∫cos(x)² dx = 8/2 ∫(1 + cos(2x))/2 dx = 4(x + 1/2 sin(2x)) + C

Substituting back u = cos(x), we obtain:

∫8(1-tan²(x)/sec²(x)) dx = 8 tan(x) + C

where C is the constant of integration.

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The complete equation of two squares to find the product of 22 and 18 is (22 + 18)(22 - 18) = 396

When we can interpret an expression as the difference of two perfect squares, i.e. a2-b2, we can factor it as (a+b)(a-b).

To use the difference of two squares to find the product of 22 and 18:

First, find the average of the two numbers:

(22 + 18) ÷ 2 = 20

Then, find the difference between the two numbers:

22 - 18 = 4

Now we can write:

(20 + 4)(20 - 4) = 24 × 16 = 384

But we need to add the extra 12 to get 396:

(20 + 4)(20 - 4) + 12 = 396

So the completed equation is:

(22 + 18)(22 - 18) = 396

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When dealing with absolute value expressions, we must consider both the positive and negative values of the argument.

In this case, we are asked to simplify[tex]|x÷3| if x<0[/tex], which means that x is a negative number.

To simplify this expression, we must first evaluate x÷3, which gives us a negative number divided by a positive number, resulting in a negative quotient.

However, since we are only interested in the absolute value of this quotient, we must ignore the negative sign and write the expression as:

[tex]|x÷3| = -(x÷3)[/tex]

Note that the negative sign in front of the expression serves to cancel out the negative sign of the quotient, thus giving us a positive result.

Therefore, the simplified expression for[tex]|x÷3| if x<0 is -(x÷3)[/tex]. This expression can be used to evaluate the value of |x÷3| for any negative value of x, by simply plugging in the corresponding value for x.

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It would take Jose approximately 3232 minutes (or about 53.87 hours) to read 2020 pages at the same rate of 2.5 pages every four minutes.

To determine how long it would take Jose to read 2020  runners at the same rate of2.5  runners every four  twinkles, we can use a proportion.   Let x be the number of  twinkles it would take Jose to read 2020  runners. also, we can set up the following proportion:

2.5 pages / 4 minutes = 2020 pages / x minutes

To solve for x, we can cross-multiply and simplify:

2.5 pages * x minutes = 4 minutes * 2020 pages

2.5x = 8080

x = 8080 / 2.5

x = 3232

Therefore, it would take Jose approximately 3232 minutes  to read 2020 pages at the same rate of 2.5 pages every four minutes.

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The minimum value of a - b is around 2.2 with a = 8.4 rounded to one decimal place and b = 6.19 rounded to two decimal places.

We must first subtract b (lower bound) from a (upper bound) to determine the least value of a - b, which is equal to 8.4 - 6.19 = 2.21. 2.21 is the difference between a and b. However, the question requests that we round off this number to the nearest tenth.

We remove the first decimal point because 1 is

less than 5, giving us 2.2. Hence the minimum value of a -b to the nearest decimal is found to be 2.2.

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The complete question is:

Given a = 8.4 rounded to 1 decimal place and b = 6.19 rounded to 2 decimal places, find the minimum value of a - b rounded to 1 decimal place.

Answer:

x=3, Adding and dividing. (Im not too sure how to answer that question, Are there some options that you learned in class?)

Step-by-step explanation:

4x-5=7

+5       +5

4x=12

/4      /4

x=3

The only line on the graph with a unit rate less than 2 is the horizontal line passing through y=8.

To identify the unit rates on the graph, we need to find the slope of the line connecting each pair of points. We can use the formula:

slope = (change in y) / (change in x)

For example, the slope between the first two points (7,8) and (14,16) is:

slope = (16-8) / (14-7) = 8/7

Similarly, we can find the slopes for the other pairs of points:

- between (7,8) and (21,24): slope = (24-8) / (21-7) = 16/14 = 8/7
- between (14,16) and (21,24): slope = (24-16) / (21-14) = 8/7

Notice that all three slopes are equal, which means the graph represents a line with a constant unit rate of 8/7.

To find lines with unit rates less than 2, we need to look for steeper lines on the graph. Any line with a slope greater than 2/8 (or 1/4) will have a unit rate greater than 2.

One way to see this is to note that a slope of 2/8 means that for every 2 units of increase in y, there is 8 units of increase in x. This is equivalent to saying that the unit rate is 2/8 = 1/4. If the slope is greater than 2/8, then the unit rate is greater than 1/4, and therefore greater than 2.

Looking at the graph, we can see that the steepest line has a slope of 2/3, which means it has a unit rate of 2/3. Therefore, any line with a slope greater than 2/3 will have a unit rate greater than 2, and any line with a slope less than 2/3 will have a unit rate less than 2.

To summarize:

- The graph represents a line with a constant unit rate of 8/7.
- Any line with a slope greater than 2/3 has a unit rate greater than 2.
- Any line with a slope less than 2/3 has a unit rate less than 2.

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About 10,676 gallons of paint were used to paint the 157 newly built homes in the subdivision, assuming each house required exactly 68 gallons of paint and 13 paint brushes.

The statement problem states that there are 157 newly built homes in a subdivision, and that 68 gallons of paint and 13 paint brushes were used for each house. This means that each house required 68 gallons of paint and 13 paint brushes.

To find the total amount of paint used for all 157 houses, we need to multiply the amount of paint used per house (68 gallons) by the number of houses (157):

Total amount of paint = 68 gallons/house x 157 houses

Total amount of paint = 10,676 gallons

Therefore, approximately 10,676 gallons of paint were used for the 157 newly built homes in the subdivision.

It's worth noting that this calculation assumes that each house required exactly 68 gallons of paint and 13 paint brushes. In reality, there may be some variation in the amount of paint and brushes used for each house, so the actual total may be slightly different.

However, this calculation provides a reasonable estimate of the total amount of paint used

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The exact length of the curve is 112/3(√3 + 1), or approximately 96.666.

To find the exact length of the curve, we can use the formula for arc length:

L = ∫a^b √(1 + [f'(x)]²) dx

where f(x) = (x² - 4)^(3/2)/6, 5 ≤ x ≤ 9.

First, we find f'(x):

f'(x) = 3x(x² - 4)^(1/2)/12 = x(x² - 4)^(1/2)/4

Then we substitute f'(x) into the formula for arc length:

L = ∫5^9 √(1 + [x(x² - 4)^(1/2)/4]²) dx

L = ∫5^9 √(1 + x²(x² - 4)/16) dx

L = ∫5^9 √(16 + 16x²(x² - 4)/16) dx

L = ∫5^9 √(16x² + x^4 - 4x²) dx

L = ∫5^9 √(x^4 + 12x²) dx

L = ∫5^9 x²√(x^2 + 12) dx

We can use the substitution u = x^2 + 12, which gives du/dx = 2x and dx = du/2x, to simplify the integral:

L = (1/2)∫37^93 √u du

L = (1/2) [(2/3)u^(3/2)]_37^93

L = (1/3)[(125 + 108√3) - (13 + 36√3)]

L = (1/3)(112√3 + 112)

L = 112/3(√3 + 1)

Therefore, the exact length of the curve is 112/3(√3 + 1), or approximately 96.666.

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