If y, p and q vary jointly and p is 14 when y and q are equal to 2, determine q when p and y are equal to 7

Answer:

1. First simplify the left-hand side of the equation using the trigonometric                      identity

sin(2t)cos(t) - cos(2t)sin(t) = sin(2t - t) = sin(t)

2. That way the equation becomes sin(t) = 0.

3. The solutions to this equation are t = kπ for all integers k.

4. Therefore, the general solution to the original equation is:

                   t = kπ or t = π/2 + kπ, where k is an integer.

Answer:

78% of sixth-graders picked a field trip to a museum.

Step-by-step explanation:

39 out of 50 kids picked the museum field trip. This is 39/50. We can change this like so:

39/50 × 2/2

= 78/100

78/100 is 78% (because percent literally means "per hundred)

Another way is to just divide. 39/50 means 39 ÷ 50.

39 ÷ 50 is .78 then times by 100 to change to a percent. This works for all kinds of fractions.

78% of sixth graders at Ayana's school selected the museum field trip.

The directional derivative of the function f(x,y) in the direction of the vector v at the point (0,1/3) is:

Duf(0,1/3) = ∇f(0,1/3) · u = [0, 3e/2] · [-5/13, 12/13] = (3e/2)(12/13) ≈ 1.38

To find the directional derivative of the function f(x, y) = 3e^x sin(y) at the point (0, 1/3) in the direction of the vector v = (-5, 12), we first need to calculate the gradient of the function, which is a vector containing the partial derivatives with respect to x and y.

The partial derivative with respect to x:
∂f/∂x = 3eˣ sin(y)

At point (0, 1/3), ∂f/∂x = 3e⁰ sin(1/3) = 3 sin(1/3)

The partial derivative with respect to y:
∂f/∂y = 3eˣ cos(y)

we first need to find the gradient of f at that point:
∇f = [∂f/∂x, ∂f/∂y] = [3ex sin(y), 3ex cos(y)]

Evaluated at (0,1/3), we get:
∇f(0,1/3) = [0, 3e/2

At point (0, 1/3), ∂f/∂y = 3e⁰ cos(1/3) = 3 cos(1/3)
So the gradient vector is ∇f = (3 sin(1/3), 3 cos(1/3)).

Next, we need to normalize the direction vector v:
|v| = √((-5)² + (12)²) = 13

Normalized vector v: (-5/13, 12/13)

Finally, we calculate the directional derivative (D_vf) as the dot product of the gradient vector and the normalized direction vector:
D(vf)= ∇f • (-5/13, 12/13) = (3 sin(1/3) × (-5/13)) + (3 cos(1/3) × (12/13))
D(vf) = (-15/13) sin(1/3) + (36/13) cos(1/3)
That is the directional derivative of the function at the given point in the direction of the vector v

Duf(0,1/3) = ∇f(0,1/3) · u = [0, 3e/2] · [-5/13, 12/13] = (3e/2)(12/13) ≈ 1.38

Therefore, the directional derivative of f(x, y) in the direction of v at the point (0,1/3) is approximately 1.38.

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The total distance travelled by dust in the given time of 4.0 minutes at the spin rate of 500rpm is equal to 26,400cm.

Spin rate of the cd is equal to 500rpm

distance of piece of dust from the center = 2.1 cm

Distance traveled by a point on the CD that is 2.1 cm from the center in one revolution =  circumference of the circle with a radius of 2.1 cm,

C = 2πr

   = 2π(2.1 cm)

   ≈ 13.2 cm

Distance traveled by the dust in one revolution is 13.2 cm.

Calculate the number of revolutions that the CD makes in 4.0 minutes.

  1 minute = 60 seconds

⇒ 4.0 minutes = 4.0 x 60

⇒4.0 minutes = 240 seconds

The CD rotates at a rate of 500 revolutions per minute,

In 4.0 minutes it will rotate,

500 x 4.0

= 2000 times.

The total distance traveled by the dust in 4.0 minutes is,

distance traveled per revolution x number of revolutions

= 13.2 cm/rev x 2000 rev

= 26,400 cm

Therefore, the total distance traveled by the dust in 4.0 minutes is 26,400 cm.

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(a) Miguel had the greatest spread in training times.

(b) Adam had the least spread in the middle 50% of training times.

(c) Miguel's training times had a greater range, indicating more variability, while Adam's training times were more consistent and tightly grouped.

(a) To determine which person had the greatest spread, we need to compare the range or variability of their training times. By observing the given data, we can see that Adam's training times range from 92 to 106, resulting in a spread of 14. On the other hand, Miguel's training times range from 85 to 105, resulting in a spread of 20. Therefore, Miguel had the greatest spread of training times.

(b) To determine which person had the least spread in the middle 50% of training times, we need to compare the interquartile range (IQR). By calculating the IQR, we find that Adam's IQR is 9 (from the 25th to the 75th percentile), whereas Miguel's IQR is 7. Since Adam's IQR is greater, it means Miguel had the least spread in the middle 50% of training times.

(c) The answers to parts (a) and (b) indicate that while Miguel had a greater spread of training times overall, Adam's training times had a greater spread in the middle 50%. This suggests that Adam's training times were more concentrated around the median, while Miguel's training times were more spread out.

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a) All of the training times of Adam had the greatest spread.

(b) The middle 50% of the training times of Adam had the least spread.

(c) The answers to Parts (a) and (b) tell us that Adam's training performance may be more stable within that middle 50%, while Miguel's performance is more variable.

(a) Adam's training times had the greatest spread.

To determine this, we can calculate the range of the data sets. For Adam, the range is 106-92 = 14 seconds, while for Miguel, the range is 105-85 = 20 seconds. However, a better measure of spread is the interquartile range (IQR), which focuses on the middle 50% of the data. For Adam, the IQR is 101-97 = 4 seconds, while for Miguel, the IQR is 96-89 = 7 seconds. In both cases, Miguel's data has a greater spread.

(b) Adam's training times had the least spread for the middle 50% of the data. This is demonstrated by the IQR, as mentioned above. For Adam, the IQR is 4 seconds, while for Miguel, it is 7 seconds.

(c) The answers to Parts 2(a) and 2(b) tell us that while Miguel's overall training times have a greater spread, the middle 50% of Adam's training times are more consistent, with less variation. This suggests that Adam's training performance may be more stable within that middle 50%, while Miguel's performance is more variable.

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The approximate area of the regular decagon, rounded to the nearest hundredth, is 190.78 square units.

To find the area of a regular decagon with an apothem of 6.2 units, we can use the formula:

Area = (1/2) × apothem × perimeter

To find "s", we can use the fact that a regular decagon can be divided into 10 congruent triangles, where each triangle has an interior angle of 144 degrees. We can use trigonometry to find the length of the side "s" using one of these triangles:

Now we can find the perimeter of the decagon:

Finally, we can substitute the apothem and perimeter into the formula to find the area:

Rounding to the nearest hundredth, the area of the regular decagon is approximately 190.78 square units.

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Ethan's monthly salary is $1450.

Ethan's monthly salary, we can use the debt to income ratio formula, which is calculated by dividing monthly debt expenses by monthly income.

Given:

Monthly expenses = $1160

Debt to income ratio = 0.8

Let's assume Ethan's monthly salary as S.

We can set up the equation using the debt to income ratio formula:

Debt to income ratio = Monthly expenses / Monthly income

0.8 = $1160 / S

To solve for S (monthly salary), we can rearrange the equation:

S = $1160 / 0.8

Dividing $1160 by 0.8 gives us:

S ≈ $1450

Therefore, Ethan's monthly salary is approximately $1450.

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We can assign each participant the real drink or placebo by assigning a drink based on the next random digit.

One method for assigning participants their drink is to utilize a basic rule predicated on the oddness or evenness of specific digits. In this scenario, you may reserve “odd” numbers strictly for real drinks and “even” numbers exclusively for placebos.

To execute this procedure, we will begin at the left-hand side of our precomputed list of randomized digits and work to identify the subsequent digit in order to assign each participant with their corresponding drink:

Assign numbers to each of the houses in every subdivision, ranging from 1 through to 100. Utilize an online random number generator or a table containing random numbers within the range of 1 and 100 to construct a list of such randomly generated figures.

Choose the initial 20 unique figures from said list for both subdivisions selectively. Proceed with visiting those particular homes denoted by these chosen numbers, then examine and test their water sources accordingly.

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The area of the shaded region is (9/500)π.

To find the area shaded below in circle K, we first need to find the radius of the circle.

Let O be the center of the circle, and let N be the midpoint of segment LM. We can draw a radius ON to segment LM such that it is perpendicular to LM, and then draw another radius OL to point L. This forms a right triangle LON with the hypotenuse equal to the radius of circle K.

Since segment LM is given to have a length of 11/9π, we can find the length of LN by dividing it in half:

LN = (11/9π)/2 = 11/18π

We can then use trigonometry to find the length of OL:

sin(55°) = OL / LN

OL = LN sin(55°)

OL = (11/18π) sin(55°)

Next, we can use the Pythagorean theorem to find the length of ON:

ON² = OL² + LN²

ON² = [(11/18π) sin(55°)]² + [11/18π]²

ON ≈ 1.022

Therefore, the radius of circle K is approximately 1.022.

The area of the shaded region can now be found by subtracting the area of sector LOM from the area of triangle LON:

Area of sector LOM = (110/360)π(1.022)² ≈ 0.317π

Area of triangle LON = (1/2)(11/18π)(1.022) ≈ 0.326π

Area of shaded region = (0.326π) - (0.317π) = (9/500)π

So the area of the shaded region is (9/500)π.

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The firm should produce 3.5 units to maximize profit, but the maximum profit is -$300, indicating the firm is operating at a loss.

To find the units of production that maximize profit, we need to first find the profit function by subtracting the cost function from the revenue function:

Profit = Revenue - Cost = R - C = 1400x - (700 + 1200x) = 200x - 700

Now, to find the units of production that maximize profit, we need to find the value of x that maximizes the profit function. We can do this by taking the derivative of the profit function with respect to x and setting it equal to zero:

d(Profit)/dx = 200 - 0 = 0

Solving for x, we get:

x = 3.5

Therefore, the firm should produce 3.5 units to maximize its profit.

To find the maximum profit, we can substitute the value of x back into the profit function:

Profit = 200x - 700 = 200(3.5) - 700 = -300

So the maximum profit is -$300, which means the firm is operating at a loss. This suggests that the firm should re-evaluate its production costs and revenue strategies to try and reduce costs or increase revenue in order to achieve a positive profit.

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14.33% of the lifetime cost of Hillary's laptop was interest.

Since Hillary paid off her laptop in two and a half years, and kept it for six years, we need to calculate the compound interest over six years. Accounting for two leap years, there were 365 * 6 + 2 = 2192 days over the period that Hillary kept the laptop. Therefore, the total cost of electricity over that period was 2192 * 0.27 = $592.64.

Plugging in the values, we get:

A = 804 * (1 + 0.1127/12)³⁰= 1003.94

Hillary paid $1003.94 for her laptop, including interest. Subtracting the original cost of the laptop, we get:

Interest = 1003.94 - 804 = 199.94

So Hillary paid $199.94 in interest on her credit card over two and a half years. To calculate what percentage of the lifetime cost of the laptop was interest, we need to divide the interest paid by the total cost of the laptop and electricity:

Lifetime cost = 804 + 592.64 = 1396.64

Percentage of lifetime cost that was interest = (199.94 / 1396.64) * 100% = 14.33%

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Based on the first survey results, we can predict that around 47 people in the second survey of 60 people will prefer cat food A over cat food B.

Based on the results of the first survey, 188 out of 240 people preferred cat food A over cat food B. To predict the preference for cat food A in the second survey, we can calculate the proportion of people who preferred cat food A in the first survey and apply it to the sample size of the second survey.

First, find the proportion of people preferring cat food A in the first survey:

Proportion = (Number of people preferring cat food A) / (Total number of people surveyed)
Proportion = 188 / 240
Proportion ≈ 0.7833

Now, apply this proportion to the second survey's sample size of 60 people:

Predicted preference = Proportion × (Sample size of the second survey)
Predicted preference = 0.7833 × 60
Predicted preference ≈ 47

Therefore, based on the first survey results, we can predict that around 47 people in the second survey of 60 people will prefer cat food A over cat food B.

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The expected number of tosses until two players get different results when simultaneously tossing a coin each with a chance of heads p is (1-2p)/(2p-2p²).

The probability that both players get the same result (either both heads or both tails) on any given toss is p² + (1-p)² = 2p² - 2p + 1. The probability that they get different results (one head and one tail) is therefore 1 - (2p² - 2p + 1) = 2p - 2p².

Let E be the expected number of tosses until they end up with different results. If they get different results on the first toss, the game ends after 1 toss.

Otherwise, they have to repeat the process again, and the expected number of tosses is increased by 1. Therefore, we can express E in terms of the probabilities of getting different or same results on the first toss

E = (2p - 2p²)1 + (1 - 2p + 2p²)(1 + E)

Simplifying, we get

E = 1 + 2pE - 2p + 2p²

Rearranging and solving for E, we get

E = (1 - 2p) / (2p - 2p²)

Therefore, the expected number of tosses before the players end up with different results is (1 - 2p) / (2p - 2p²).

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First algebraic expression: x + 2. Second algebraic expression: 6x + 12.

We can represent Harvey's winnings using algebraic expressions.

Let's use the variable 'x' to represent the amount Harvey won on the scratch-and-win ticket. Harvey then won a $2 bonus, so we add 2 to 'x':

1) x + 2

Next, Harvey won a "triple your winnings" ticket, so we need to multiply the current winnings by 3:

2) 3(x + 2)

Finally, as the 100th customer, Harvey's total winnings were doubled:

3) 2 * 3(x + 2)

So, the two algebraic expressions to describe Harvey's winnings are:

1) x + 2 (initial winnings with the $2 bonus)
2) 2 * 3(x + 2) (total winnings after tripling and doubling)

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The critical point(s) of the function f(x) = x^3 + x - 3 + 2 are determined to find the x-coordinate(s) of the local minimum or local maximum.

To find the critical point(s) of the given function, we need to first find the derivative of the function and then solve for the value(s) of x that make the derivative equal to zero.

Given function: f(x) = x^3 + x - 3 + 2

Find the derivative of the function f(x) with respect to x.

f'(x) = 3x^2 + 1

Set the derivative f'(x) equal to zero and solve for x.

3x^2 + 1 = 0

Subtract 1 from both sides of the equation.

3x^2 = -1

Divide both sides of the equation by 3.

x^2 = -1/3

Take the square root of both sides of the equation.

x = ±√(-1/3)

Since the square root of a negative number is not a real number, the function f(x) does not have any real critical points. Therefore, the critical point(s) for the function f(x) = x^3 + x - 3 + 2 is DNE (Does Not Exist) in terms of real numbers.

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Answer:

9

Step-by-step explanation:

First, let's move the variables to one side and the numbers to the other side:

[tex]\frac{2}{3}b+5=20-b\\[/tex]

subtract 5 from both sides:

[tex]\frac{2}{3}b=15-b\\[/tex]

add b to both sides:

[tex]\\1\frac{2}{3}b=15\\[/tex]

divide both sides by [tex]1\frac{2}{3}[/tex]:

[tex]b=9[/tex]

Hope this helps :)

The value of T is 428°C.

To solve the problem, we can start by substituting the given value of R = 13946 into the equation R = 10000 + (4124 x 10^-2)T – (1779 x 10^-5)T^2 and solving for T. This gives us a quadratic equation in T which can be solved using the quadratic formula.

After simplifying, we get T = 427.67°C or T = -88.22°C. However, we know that the thermometer is only used if T ≤ 600°C, so the only valid solution is T = 427.67°C.Therefore, the temperature corresponding to a resistance of 13946 ohms is approximately 428°C.

It's important to note that this assumes the thermometer is operating within its specified range and that the resistance-temperature relationship remains linear over the given temperature range.

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Answer:

Step-by-step explanation:

You're going to want to break up the shape into three parts, two triangles, and the rectangle.

Starting with the left-most triangle: A=(L*W)/2

The length is 4ft and the width is 3ft, multiply and divide by 2 to get:  A=6 square feet.

Do the same with the second triangle on the bottom left (L=2ft, W=2ft) to get A=2 square feet.

Now the rectangle, A=L*W and total length is 10ft (8ft+2ft) and the width is 3ft. Multiply these values to get A=30 square feet.

Last step: add up all three areas for the total area of the entire shape, 6+2+30=38.

Area= 38 square feet.

la probabilidad de que Antawn Jamison anote sus siguientes 9 tiros libres es del 7.33%.

What is probability?

By simply dividing the favorable number of possibilities by the entire number of possible outcomes, the probability of an occurrence can be determined using the probability formula. Because the favorable number of outcomes can never exceed the entire number of outcomes, the chance of an event occurring might range from 0 to 1.

La probabilidad de que anote un tiro libre es del 73%, lo que significa que la probabilidad de que falle es del 27%.

La probabilidad de que anote sus próximos 9 tiros libres es:

0.73 x 0.73 x 0.73 x 0.73 x 0.73 x 0.73 x 0.73 x 0.73 x 0.73 = 0.0733

Por lo tanto, la probabilidad de que Antawn Jamison anote sus siguientes 9 tiros libres es del 7.33%.

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To find the value of x when (fog)(x) = -8, we first need to find the composition of f and g, which is given by (fog)(x) = f(g(x)). To do this, we substitute g(x) into the expression for f(x) and simplify:

f(g(x)) = f(1) when g(x) = 1

f(g(x)) = f(-2) when g(x) = -2

f(g(x)) = f(3) when g(x) = 3

f(g(x)) = f(-4) when g(x) = -4

f(g(x)) = f(4) when g(x) = 4

There is no value of x for which (fog)(x) = -8.

Using the table given in the question, we can find the values of f(g(x)) for each possible value of g(x):

f(g(x)) = f(1) = -2

f(g(x)) = f(-2) = 0

f(g(x)) = f(3) = 32

f(g(x)) = f(-4) = 8

f(g(x)) = f(4) = 4

Therefore, (fog)(x) = -8 is not possible. The closest value we can get to -8 is by setting g(x) = -4, which gives f(g(x)) = f(-4) = 8. Thus, there is no value of x for which (fog)(x) = -8.

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Answer:

i think the answer is 20.25

Step-by-step explanation:

The combinations of side lengths that would NOT form a triangle with vertices X, Y, and Z is 7 mm , YZ = 14 mm , XZ = 25 mm.

option A.

The lengths of triangle are determined base a given set of rules;

let a, b, and c be the side lengths of a triangle;

Based on the rules of side lengths of triangles, the sum of length a and b must be greater than c, or the sum of a and c must be greater than b or the sum of b and c must be greater than a.

For option A;

7 mm + 14 mm < 25 mm (this cannot be)

For option B;

11 mm + 18 mm > 21 mm (this will work)

For option C;

11 mm + 14 mm > 21 mm (this will work)

For option D;

7 mm + 14 mm > 17 mm (this will work)

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Answer:40-45 percent

Step-by-step explanation:

explenation

Answer:

Sure. Here are the steps on how to solve for x^3 - 1/x^3:

1. **Cube both sides of the equation x - 1/x = 3.** This will give us the equation x^3 - 3x + 1/x^3 = 27.

2. **Subtract 1 from both sides of the equation.** This will give us the equation x^3 - 1/x^3 = 26.

3. **The answer is 26.**

Here is the solution in detail:

1. **Cube both sides of the equation x - 1/x = 3.**

```

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

```

```

x^3 - 3x + 1/x^3 = 27

```

2. **Subtract 1 from both sides of the equation.**

```

x^3 - 1/x^3 - 1 = 27 - 1

```

```

x^3 - 1/x^3 = 26

```

3. The answer is 26.

The finance charge on a credit card account with a balance of $660.30 and an APR of 6.2% is $3.41.

To calculate the finance charge on a credit card account with a balance of $660.30 and an APR of 6.2%. Here's a step-by-step explanation:

1. Convert the APR (Annual Percentage Rate) to a decimal by dividing it by 100: 6.2 / 100 = 0.062

2. Divide the APR decimal by 12 to find the monthly interest rate: 0.062 / 12 = 0.005167

3. Multiply the credit card balance by the monthly interest rate: $660.30 * 0.005167 = $3.41

The finance charge on a credit card account with a balance of $660.30 and an Annual Percentage Rate (APR) of 6.2% is determined to be $3.41. This finance charge represents the cost of borrowing on the credit card and is calculated based on the outstanding balance and the interest rate.

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If the probability of picking out a basketball card is 2/5, and the probability of picking out a football card is 1/3, the probability of Justice randomly picking out a soccer card is 4/15.

To find the probability of Justice picking out a soccer card, we need to know the total probability of all three types of cards adding up to 1. Since there are only three types of cards, we can subtract the probability of picking a basketball card and the probability of picking a football card from 1 to find the probability of picking a soccer card.

Let P(S) be the probability of picking a soccer card.

We know that P(B) = 2/5 and P(F) = 1/3.

Therefore, the total probability of picking one of the three cards is:

P(B) + P(F) + P(S) = 1

Substituting the values we know, we get:

2/5 + 1/3 + P(S) = 1

Simplifying the equation, we get:

6/15 + 5/15 + P(S) = 1

11/15 + P(S) = 1

P(S) = 1 - 11/15

P(S) = 4/15

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To find the arc length of the curve y = -x² -5 on the interval [-1,2], we use the formula to evaluate:

L = ∫√(1 + (dy/dx)²) dx

where dy/dx is the derivative of y with respect to x.

First, we find dy/dx:

dy/dx = -2x

Next, we substitute dy/dx into the formula and simplify:

L = ∫√(1 + (-2x)²) dx
L = ∫√(1 + 4x²) dx

To evaluate this integral, we can use a trigonometric substitution. Let x = (1/2)tanθ, then dx = (1/2)sec²θ dθ. Substituting, we get:

L = ∫√(1 + 4(1/2)²tan²θ)(1/2)sec²θ dθ
L = (1/2)∫sec³θ dθ

To integrate sec³θ, we use integration by parts:

u = secθ, du/dθ = secθ tanθ
dv/dθ = sec²θ, v = tanθ

∫sec³θ dθ = secθ tanθ - ∫tan²θ secθ dθ
= secθ tanθ - ∫secθ dθ + ∫sec³θ dθ

Rearranging, we get:

2∫sec³θ dθ = secθ tanθ + ln|secθ + tanθ|

Therefore:

L = (1/2)(secθ tanθ + ln|secθ + tanθ|) + C

To evaluate L on the interval [-1,2], we need to find θ when x = -1 and x = 2. Using the substitution x = (1/2)tanθ:

When x = -1, θ = -π/4
When x = 2, θ = π/3

Substituting these values into the equation for L and simplifying, we get:

L = (1/2)(2√2 + ln(3 + 2√3) + π/4)

Therefore, the integral that gives the arc length of the curve y = -x² -5 on the interval [-1,2] is:

L = (1/2)(2√2 + ln(3 + 2√3) + π/4)

Note: If technology is used to evaluate or approximate the integral, the answer may differ slightly due to rounding errors.

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The height of the triangle is 8 inches and the width is 7 inches.

Let's start by using the formula for the area of a triangle:

A = (1/2)bh

where A is the area of the triangle, b is the base, and h is the height.

We are given that the area of the triangle is 28 square inches, so we can write:

28 = (1/2)bh

Next, we are given that the height h is six less than twice the width w. In other words:

h = 2w - 6

Now we can substitute this expression for h into the formula for the area:

28 = (1/2)bw(2w - 6)

Simplifying this equation, we get:

56 = bw(2w - 6)

28 = w(w - 3)

w^2 - 3w - 28 = 0

We can solve this quadratic equation using the quadratic formula:

w = [3 ± √ ([tex]3^2[/tex] - 4(1)(-28))] / 2

w = [3 ± √ (121)] / 2

w = (3 + 11) / 2 or w = (3 - 11) / 2

w = 7 or w = -4

Since a negative width doesn't make sense in this context, we can ignore the second solution and conclude that the width of the triangle is 7 inches.

Now we can use the expression for h in terms of w to find the height:

h = 2w - 6

h = 2(7) - 6

h = 8

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Answer:

The perimeter of rectangle ABCD can be calculated by adding up the lengths of its sides. Using the distance formula, we can find that AB has a length of 2 units, BC has a length of 2 units, CD has a length of 2 units, and AD has a length of 4 units. Therefore, the perimeter of rectangle ABCD is 10 units.

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