Item #1 - Which set of data points could be modeled by a
decreasing linear function?
A. {(0, 0), (1, 8), (2, 15), (3, 22), (4, 30)}
B. {(0, 5), (1, 6), (2, 10), (3, 16), (4, 28)}
C. {(0, 50), (1, 42), (2, 33), (3, 25), (4, 16)}
D. {(0, 64), (1, 60), (2, 52), (3, 39), (4, 22)}

The derivative of the function f(x) = 2 \sqrt(x) / (8x² + 3x - 9) evaluated at x = 4.

To find f'(x), we need to differentiate the given function f(x) using the power rule and the chain rule of differentiation.

First, we can rewrite the function f(x) as:

f(x) = 2x^{1/2} / (8x² + 3x - 9)

Next, we can differentiate f(x) with respect to x:

f'(x) = d/dx [2x^{1/2} / (8x² + 3x - 9)]

Using the quotient rule of differentiation, we have:

f'(x) = [ (8x² + 3x - 9) d/dx [2x^{1/2}] - 2x^{1/2} d/dx [8x² + 3x - 9] ] / (8x² + 3x - 9)²

Applying the power rule of differentiation, we have:

f'(x) = [ (8x² + 3x - 9)(1/2) - 2x{1/2}(16x + 3) ] / (8x² + 3x - 9)²

Now we can evaluate f'(x) at x = 4 by substituting x = 4 into the expression for f'(x):

f'(4) = [ (8(4)² + 3(4) - 9)(1/2) - 2(4)^(1/2)(16(4) + 3) ] / (8(4)² + 3(4) - 9)²

f'(4) = [ (128 + 12 - 9)(1/2) - 2(4)^(1/2)(67) ] / (128 + 12 - 9)^2

f'(4) = [ 131^(1/2) - 2(4)^(1/2)(67) ] / 12167

Therefore, f'(4) = [ 131^(1/2) - 134(2)^(1/2) ] / 12167.

This is the value of the derivative of f(x) at x = 4.

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If in Angle STU, the measure of U=90°, the measure of S=31°, and TU = 77 feet, then the length of US to the nearest tenth of a foot is approximately 39.4 feet.

In angle STU, we have a right triangle with U=90°, S=31°, and TU=77 feet. To find the length of US, we can use the sine function:

sin(S) = opposite side (US) / hypotenuse (TU)

sin(31°) = US / 77 feet

To find the length of US, multiply both sides by 77 feet:

US = 77 feet * sin(31°)

US ≈ 39.4 feet

Therefore, the length of US to the nearest tenth of a foot is approximately 39.4 feet.

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A relative frequency of 0.56 means that out of the total number of observations in a given sample or population, 56% of those observations belong to a particular category or have a certain characteristic.

In other words, it is the proportion or fraction of the observations that fall into that particular category or have that characteristic, relative to the total number of observations. For example, if we had a sample of 100 people and 56 of them had brown hair, then the relative frequency of brown hair would be 0.56 or 56%.

To calculate relative frequency, you divide the frequency of a specific event or category by the total number of observations. In this case, the specific event or category occurs 56% as often as the total events or categories observed.

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Answer:

a and b

Step-by-step explanation:

Comets could be visible from Earth when they are most likely to fall down into earth

Step-by-step explanation:

To determine the appropriate fraction for the line plot, we need to find the greatest common factor (GCF) of all the pencil lengths, and then express each length as an equivalent fraction with the GCF as the denominator.

The GCF of the pencil lengths is 1. Therefore, we can simply express each pencil length as an equivalent fraction with 1 as the denominator:

3/1, 7/1, 8/1, 5/1, 1/1, 4/1, 6/1, 8/1, 4/1, 2/1, 5/1

Now, we can see that the smallest unit increment we can use on the line plot is 1/8. This is because 1/8 is the smallest fraction that can represent all of the pencil lengths above the label (5/8, which is equivalent to 10/16).

Therefore, the students should use 1/8 as the fraction for the line plot.

Approximate amount of dough that is remaining. Is  (length - 2r)(width - 3r) - 6πr^2.

Without the size of the original sheet of dough or the size of the circles cut from it, it's not possible to give an exact expression. However, assuming that each circle has the same radius of 'r' and the original sheet of dough was a rectangle, we can write an expression in factored form for the remaining area of the dough:

Remaining area of dough = (Area of original rectangle) - 6(Area of circle)

= (length x width) - 6(πr^2)

= (length - 2r)(width - 3r) - 6πr^2

Whether there is enough dough for another circle would depend on the size of the circles and the original sheet of dough.

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Answer:

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

The given point (0, 5) represents the y-intercept and we have a slope of -1.

It translates as:

By substituting we get equation:

This is option C.

Answer:

.70p = £420, so p = £600

The original price of the bracelet is £600.

The original price of the bracelet was £600.

To find the original price of the bracelet, we need to use the information that the current price is 70% of the original price. We can use algebra to solve for the original price:

Let X be the original price of the bracelet.

70% of X is equal to £420.

We can write this as:

0.7X = £420

To solve for X, we can divide both sides of the equation by 0.7:

X = £420 ÷ 0.7

Evaluating the right-hand side gives us:

X = £600

Therefore, the original price of the bracelet was £600.

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The equation of the hyperbola is -(x + 5)²/71 + (y - 2)² = -71

We can also find the distance between the center and the given focus, which is the distance between (-5, 2) and (-5 - √29, 2):

d = |-5 - (-5 - √29)| = √29

Substituting in the known values, we get:

c² = a² + b²

(√29)² = (10)² + b²

29 = 100 + b²

b² = -71

(x - h)²/a² - (y - k)²/b² = 1

where (h, k) is the center of the hyperbola.

Substituting in the known values, we get:

(x + 5)²/100 - (y - 2)²/-71 = 1

Multiplying both sides by -71, we get:

-(x + 5)²/71 + (y - 2)²/1 = -71/1

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Answer:

Yes

Step-by-step explanation:

Yes, because its graph represents a straight line

Entry debits the Depreciation Expense account for $7,000 and credits the Accumulated Depreciation account for the same amount, reflecting the decrease in the value of the machine over time.

To calculate  deprecation using the straight- line  system, we need to abate the residual value from the  original cost of the machine and  also divide the result by the estimated useful life. Using the given values, we have  

Cost of machine = $ 39,000

Residual value = $ 4,000  

Depreciable cost = $ 35,000($ 39,000-$ 4,000)

Estimated useful life =  5 times  

To calculate the periodic  deprecation  expenditure, we divide the depreciable cost by the estimated useful life  

Periodic  deprecation  expenditure = $ 7,000($ 35,000 ÷ 5)

Depreciation Expense $7,000

Accumulated Depreciation $7,000

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The journal entry is as given in figure:

Shown that if f has no local minima nor local maxima on i, then f is either increasing or decreasing on i.

Since f has no local minima nor local maxima on i, it means that for any point x in i, either f is increasing or decreasing in a small interval around x. In other words, either f'(x) > 0 or f'(x) < 0 for all x in i.

Now suppose there exist two points a and b in i such that a < b and f(a) < f(b). We want to show that f is increasing on i.

Consider the interval [a,b]. By the Mean Value Theorem, there exists a point c in (a,b) such that f'(c) = (f(b) - f(a))/(b - a). Since f(a) < f(b), we have (f(b) - f(a))/(b - a) > 0, which implies f'(c) > 0. Since f'(x) > 0 for all x in i, it follows that f is increasing on [a,c] and on [c,b]. Therefore, f is increasing on i.

A similar argument can be made if f(a) > f(b), which would imply that f is decreasing on i.

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Step-by-step explanation:

g(x)  is just f(x) shifted UP three units ...so

  g(x) = f(x) +3

The function f representing the sum of the series for x in the interval (-3, 3). Hi! The given geometric series is 1 - x/3 - x^2/9 - x^3/27...

The common ratio of the series is obtained by dividing a term by its preceding term. Let's consider the first two terms:

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

Therefore, the common ratio (r) of the series is -x/3.

For a geometric series to converge, the absolute value of the common ratio must be less than 1, i.e., |r| < 1. In this case:

|-x/3| < 1

To find the values of x for which the series converges, we need to solve the inequality:

-1 < x/3 < 1

Multiplying all sides by 3, we get:

-3 < x < 3

So, the series converges for x in the interval (-3, 3).

Now, let's determine the function f representing the sum of the series. For a converging geometric series, the sum S can be calculated using the formula:

S = a / (1 - r)

where a is the first term and r is the common ratio. In this case, a = 1 and r = -x/3. Therefore:

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

This is the function f representing the sum of the series for x in the interval (-3, 3).

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Answer:

360 cubes

Step-by-step explanation:

Answer:

Step-by-step explanation:

To find the solution set to the inequality w - 10 < 16, we can solve for w by adding 10 to both sides of the inequality:

w - 10 + 10 < 16 + 10 w < 26

This means that any number less than 26 is part of the solution set to the inequality. So, out of the given options, the number that is not part of the solution set is D. 27 because it is greater than 26.

The volume of the cylinder is 4004 cubic metres.

The diameter of the cylinder is 28 metres and the height of the cylinder is 6.5 metres.

Therefore, the volume of the cylinder can be found as follows:

Hence,

volume of a cylinder = πr²h

where

Therefore,

volume of the cylinder =  22 / 7 × 14² × 6.5

volume of the cylinder = 22 / 7 × 196 × 6.5

volume of the cylinder = 28028 / 7

volume of the cylinder = 4004 cubic metres

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Answer:

A′(−2.5, 3.75), B′(−1.25, 1.25), C′(2.5, −1.25), D′(2.5, 2.5)

Step-by-step explanation:

in the described situation you only need to multiply the coordinates by the scale factor (in our case the given 5/8)

A (-4, 6) turns into

A' (-4×5/8, 6×5/8) = A' (-2.4, 3.75)

and therefore we know already here that all the other answer options are wrong.

Taking the data into consideration, Shelby should use 50 mL of the 10% glycerin solution and 100 mL of the 40% glycerin solution, as explained below.

Let x be the amount of 10% glycerin solution and y be the amount of 40% glycerin solution that Shelby needs to use. We know that Shelby needs a total of 150 mL of the 30% glycerin solution, so we can write:

x + y = 150 (equation 1)

We also know that the concentration of glycerin in the 10% solution is 10%, and the concentration of glycerin in the 40% solution is 40%. So, the amount of glycerin in x mL of the 10% solution is 0.1x, and the amount of glycerin in y mL of the 40% solution is 0.4y. The total amount of glycerin in the 150 mL of 30% solution is 0.3(150) = 45 mL. So, we can write:

0.1x + 0.4y = 45 (equation 2)

We now have two equations with two variables. We can use substitution or elimination to solve for x and y. Here, we'll use elimination. Multiplying equation 1 by 0.1, we get:

0.1x + 0.1y = 15 (equation 3)

Subtracting equation 3 from equation 2, we get:

0.3y = 30

y = 100

Substituting y = 100 into equation 1, we get:

x + 100 = 150

x = 50

Therefore, Shelby needs to use 50 mL of the 10% glycerin solution and 100 mL of the 40% glycerin solution.

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If angle of arc EA is 112 degrees then value of arc IV is 36 degrees by  outside angles theorem

Given that Arc EA measure is One hundred twelve degrees

By Outside Angles Theorem states that the measure of an angle formed by two secants, two tangents, or a secant and a tangent from a point outside the circle is half the difference of the measures of the intercepted arcs

(112-x)/2=38

112-x=38×2

112-x=76

112-76=x

36 degrees = angle IV or x

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To find the length of AC in triangle ABC, we will use the Angle Bisector Theorem and the given information:
In triangle ABC, the bisector of angle A divides BC into segments BD and DC, with lengths of 28 units and 24 units, respectively. Given that AB has a length of 31.5 units, we want to determine the possible length of AC.

Step 1: Apply the Angle Bisector Theorem, which states that the ratio of the lengths of the sides is equal to the ratio of the lengths of the segments created by the angle bisector. In this case, we have:

AB / AC = BD / DC

Step 2: Plug in the known values:

31.5 / AC = 28 / 24

Step 3: Simplify the ratio on the right side:

31.5 / AC = 7 / 6

Step 4: Cross-multiply to solve for AC:

6 * 31.5 = 7 * AC

Step 5: Calculate the result:

189 = 7 * AC

Step 6: Divide both sides by 7 to find AC:

AC = 189 / 7

Step 7: Calculate the value of AC:

AC = 27 units

So, the length of AC in triangle ABC could be 27 units.

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The grocery store's earning income is $30,030 if they sell 200 tomatoes in one day.

We need to find how much the grocery store earns when it sells 200 tomatoes in one day. When The grocery store’s earnings in dollars can be modeled by the equation,

y = 0.75x² + 0.15x

where,

x = number of tomatoes they sell = 200

To find the earnings we need to substitute x in the equation it can be given as,

y = 0.75x² + 0.15x

y = 0.75(200)² + 0.15(200)

y = $30,030

Therefore, the grocery store's earning income is $30,030 if they sell 200 tomatoes in one day.

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Answer:

To make 3 gallons of punch, you need 2 cups of cranberry juice.

We can set up a proportion to find out how much cranberry juice is needed for 1 gallon of punch:

2 cups / 3 gallons = x cups / 1 gallon

To solve for x, we can cross-multiply:

2 cups * 1 gallon = 3 gallons * x cups

2 cups = 3x

x = 2/3 cup

Therefore, you would need 2/3 cup of cranberry juice to make 1 gallon of punch using this recipe.

The student used 30.0 mL of 3.00 M HCl to make the 50.0 mL sample of 1.80 M HCl.

To find the volume of 3.00 M HCl needed to make a 50.0 mL sample of 1.80 M HCl, we can use the equation:

M₁V₁ = M₂V₂

Where M₁ is the initial concentration, V₁ is the initial volume, M₂ is the final concentration, and V₂ is the final volume.

We are given M₁ = 3.00 M, M₂ = 1.80 M, and V₂ = 50.0 mL. We can rearrange the equation to solve for V₁:

V₁ = (M₂V₂) / M₁

V₁ = (1.80 M * 50.0 mL) / 3.00 M

V₁ = 30.0 mL

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The volume of cement in cubic feet that will be needed is 45 cubic feet.

Volume is the space occuppied by an object.

To calculate the volume of cement in cubic feet that will be needed, we use the formula below

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1

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Answer:

$264,000

Step-by-step explanation:

PV = $110,000

i = 6%

n = 15 years

Compound formula:

FV = PV (1 + i)^n

FV = 110,000 (1 + 0.06)^15

FV = 110,000 · 2.40(rounded) = $264,000

All the value of angles are,

⇒ ∠a = 118°

⇒ ∠b = 62°

⇒ ∠q = 84°

⇒ ∠v = 84°

Given that;

Two parallel lines t₁ and t₂ are shown in image.

Here, we have;

Apply the definition of corresponding angles,

∠d = 180 - 62

∠d = 118°

Hence, By definition of vertically opposite angles,

⇒ ∠a = 118°

And,

∠b = 180 - 118°

∠b = 62°

Apply the definition of corresponding angles,

∠q = 180 - 96

∠q = 84°

Hence, By definition of vertically opposite angles,

⇒ ∠v = 84°

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Answer:

Step-by-step explanation:

If there are 180 degrees in a triangle total and in this problem we know that one angle is 49 and the other is 39, we can assume that subtracting 39 and 49 from 180 will find x. In this case, x will be 92.

The point (–2, 5) is located above the point (–2, –1).

The point (1, 212) is located to the left of the point (4, 212).

The point (3, –6) is located below the point (3, –3).

The point (−212, 1) is located to the left of the point (–3, 1).

The point (312, 12) is located to the right of the point (12, 12).

The point (2, 5) is located above the point (2, –5).

a. (–2, 5) is above (–2, –1). The two points have the same x-coordinate, but different y-coordinates. Since the y-coordinate increases as you move up on the coordinate plane, the point (–2, 5) is located above the point (–2, –1).

b. (1, 212) is to the left of (4, 212). The two points have the same y-coordinate, but different x-coordinates. Since the x-coordinate increases as you move to the right on the coordinate plane, the point (1, 212) is located to the left of the point (4, 212).

c. (3, –6) is below (3, –3). The two points have the same x-coordinate, but different y-coordinates. Since the y-coordinate decreases as you move down on the coordinate plane, the point (3, –6) is located below the point (3, –3).

d. (−212, 1) is to the left of (–3, 1). The two points have the same y-coordinate, but different x-coordinates. Since the x-coordinate decreases as you move to the left on the coordinate plane, the point (−212, 1) is located to the left of the point (–3, 1).

e. (312, 12) is to the right of (12, 12). The two points have the same y-coordinate, but different x-coordinates. Since the x-coordinate increases as you move to the right on the coordinate plane, the point (312, 12) is located to the right of the point (12, 12).

f. (2, 5) is above (2, –5). The two points have the same x-coordinate, but different y-coordinates. Since the y-coordinate increases as you move up on the coordinate plane, the point (2, 5) is located above the point (2, –5).

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