Use the diagram to match the terms with the correct example.

We get:

[tex]y' = [x k_1 sin(k_1 x + k_2 y) - cos(k_1 x + k_2 y)] / [cos x - y sin x][/tex]

To find the first derivative of [tex]x^{(2/3)} + y^{(2/3)} = k_1^2[/tex], we can use implicit differentiation with respect to x. Taking the derivative of both sides, we get:

[tex](2/3)x^{(-1/3)} dx/dx + (2/3)y^{(-1/3)} dy/dx = 0[/tex]

Simplifying and solving for [tex]dy/dx[/tex], we get:

[tex]dy/dx = - (x/y)(y/x)^{(-2/3)} = - (x/y) (y/x)^{(2/3)}[/tex]

which can also be written as:

[tex]dy/dx = - (y/x)^{(1/3)}[/tex]

To find the first derivative of [tex]x cos(k_1 x + k_2 y) = y sin x[/tex], we can also use implicit differentiation with respect to x. Taking the derivative of both sides, we get:

[tex]cos(k_1 x + k_2 y) - x k_1 sin(k_1 x + k_2 y) = y \cos x[/tex]

Solving for y' (i.e., [tex]dy/dx[/tex]), we get:

[tex]y' = [x k_1 sin(k_1 x + k_2 y) - cos(k_1 x + k_2 y)] / [cos x - y sin x][/tex]

Note that we could have also solved for x' (i.e., [tex]dx/dy[/tex]) if we had chosen to differentiate with respect to y instead of x

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a. There are 1000 mice in the neighborhood initially.

b.  The population of mice never quadruple

c. After 5 weeks there are 18 mice in the neighborhood.

d. It takes 0 weeks for there to be 1000 mice.

e. The P' (5) is -96.86, indicates that after 5 weeks, the number of mice is declining at a pace of about 96.86 mice per week.

a. The initial number of mice in the neighborhood can be found by evaluating P(0):

P(0) = 2000/(1 + 10⁰/₁₀) = 2000/(1+1) = 1000

b. To find how long it takes for the population of mice to quadruple, we need to solve the equation:

P(t) = 4P(0)

2000/(1 + 10^(t/10)) = 4*1000

1 + 10^(t/10) = 1/4

10^(t/10) = -3/4

This equation has no real solutions, so the population of mice never quadruples.

c. To find how many mice are in the neighborhood after 5 weeks, we simply evaluate P(5):

P(5) = 2000/(1 + 10^(5/10)) = 2000/(1+100) = 18.18 (rounded to two decimal places)

Therefore, there are approximately 18 mice in the neighborhood after 5 weeks.

d. To find how long it takes until there are 1000 mice, we need to solve the equation:

P(t) = 1000

2000/(1 + 10^(t/10)) = 1000

1 + 10^(t/10) = 2

10^(t/10) = 1

t = 0

Therefore, there are 1000 mice in the neighborhood initially, so it takes 0 weeks for there to be 1000 mice.

e. To find P'(5), we first find the derivative of P(t):

P'(t) = -2000ln(10)/10 * 10^(t/10) / (1 + 10^(t/10))^2

Then we evaluate P'(5):

P'(5) = -2000ln(10)/10 * 10^(1/2) / (1 + 10^(1/2))^2 ≈ -96.86

This means that the population of mice is decreasing at a rate of approximately 96.86 mice per week after 5 weeks.

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Answer:

Yes they are mutually exclusive

Step-by-step explanation:

You cannot both be a rational number and an irrational number

To estimate your paycheck, a good percentage to multiply your earning by would be 0.75 or 75%. When calculating your paycheck, it's important to account for taxes, deductions, and other withholdings that may be taken out of your gross pay.  

This accounts for taxes, deductions, and other withholdings that are typically taken out of your paycheck before you receive your net pay. For example, if you earn $1,000 per pay period, multiplying by 0.75 would give you an estimated net pay of $750. However, keep in mind that this is just an estimate and your actual net pay may vary depending on your specific tax situation and other factors.

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The derivative of s with respect to t is:

ds/dt = 10(sec² t - tan t) * (tan² t - sec² t)⁴

To find the derivative of s with respect to t, we will use the chain rule and the power rule of differentiation.

Let u = (tan² t - sec² t). Then, s = u⁵.

Using the chain rule, we have:

ds/dt = (du/dt) * (ds/du)

Now, we need to find du/dt and ds/du.

Using the chain rule again, we have:

du/dt = d/dt(tan² t - sec² t) = 2tan t * sec² t - 2sec t * tan t * sec t = 2sec² t * (tan t - sec t)

To find ds/du, we can simply apply the power rule:

ds/du = 5u⁴

Substituting these into the original equation for ds/dt, we get:

ds/dt = (2sec² t * (tan t - sec t)) * (5(tan² t - sec² t)⁴)

Therefore, the derivative of s with respect to t is:

ds/dt = 10(sec² t - tan t) * (tan² t - sec² t)⁴

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Answer:

Step-by-step explanation:

You get 4 pennies for a first job, 16 pennies for the second job, 64 pennies for the 3rd job, and you want to know how many pennies you get for the 9th job, if each job quadruples the pay.

We can write the number of pennies as a power of 4:

You will get 4^9 pennies for the 9th job.

That is 262144 pennies.

<95141404393>

Answer:148

Step-by-step explanation:

Assuming he only bought milk, fish and chicken.
220+132=352
500-352=148

The range of values that represent the middle 95% of the data is from 20.18 to 47.82 or (20.18, 47.82).

A measurable speculation test is a strategy for factual deduction used to conclude whether the information within reach adequately support a specific speculation. We can make probabilistic statements about the parameters of the population thanks to hypothesis testing.

The middle 95% of a normal distribution is located within 1.96 standard deviations from the mean in both directions.

Therefore, the lower limit is:

34 - 1.96(7) = 20.18

And the upper limit is:

34 + 1.96(7) = 47.82

So the range of values that represent the middle 95% of the data is from 20.18 to 47.82 or (20.18, 47.82).

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The amount of cashews needed to be mixed with the peanuts so that the mixture will produce the same revenue as selling the nuts separately is 10 pounds.

To solve this problem, we need to use the equation:

$3(20) + 6x = 4(20 + x)$

where x is the number of pounds of cashews needed.

First, we simplify the equation by multiplying:

$60 + 6x = 80 + 4x$

Then we isolate x by subtracting 4x from both sides and subtracting 60 from both sides:

$2x = 20$

Finally, we solve for x by dividing both sides by 2:

$x = 10$

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[tex]~~~~~~ \textit{Simple Interest Earned Amount} \\\\ A=P(1+rt)\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$25000\\ r=rate\to 5.1\%\to \frac{5.1}{100}\dotfill &0.051\\ t=years\dotfill &4 \end{cases} \\\\\\ A = 25000[1+(0.051)(4)] \implies A=25000(1.204)\implies A = 30100[/tex]

the repeating decimal 0.216 can be written as the geometric series: 0.216 = 216/990.

To write the repeating decimal 0.216 as a geometric series, we first need to express it in the form of a sum of a geometric series.

The decimal 0.216 repeats every three digits, so we can break it down as follows:

0.216 = 0.2 + 0.01 + 0.006 + 0.0002 + 0.00001 + 0.000006 + ...

Now, we can write this as a sum of a geometric series with the first term (a) and the common ratio (r):

a = 0.2
r = 0.01 (because each term is 1/100 of the previous term)

Thus, the geometric series for the repeating decimal 0.216 is:

0.216 = 0.2 + 0.2(0.01) + 0.2(0.01)^2 + 0.2(0.01)^3 + ...

The formula for the sum of an infinite geometric series is S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.

Using the values for a and r, we can find the sum of the series:

S = 0.2 / (1 - 0.01) = 0.2 / 0.99 = 216/990.

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If Sin F = 3/5 , then the value of Cos D is 4/5 (option a)

Let us consider the triangle in the given question. Since we are given that Sin F = 3/5, we know that the side opposite angle F is 3 and the hypotenuse is 5. Using Pythagoras theorem, we can find the length of the adjacent side as follows:

Opposite² + Adjacent² = Hypotenuse²

3² + Adjacent² = 5²

9 + Adjacent² = 25

Adjacent² = 16

Adjacent = 4

So we have found that the length of the adjacent side is 4. Now we can use the definition of cosine to find Cos D.

Cosine is defined as the ratio of the adjacent side to the hypotenuse. Therefore,

Cos D = Adjacent/Hypotenuse = 4/5

Hence, the answer is option A) 4/5.

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All the value of of that satisfy the condition cos θ = - 1/2 ,

where 0 ≤ θ <360° are,

⇒ 120°, 240°

Given that;

Expression is,

cos θ = - 1/2

where 0 ≤ θ <360°

Since, cos θ = - 1/2

Hence, It belong in second and third quadrant.

So, cos θ = - 1/2

cos θ = 120°

cos θ = 240°

Thus, All the value of of that satisfy the condition cos θ = - 1/2 ,

where 0 ≤ θ <360° are,

⇒ 120°, 240°

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The answer is "No they are not similar". The correct option is D.

To determine if the two cuboids are similar, we need to check if their corresponding sides are proportional.

If we compare the corresponding sides of the first cuboid and the second cuboid, we get:

6/10 = 0.6

8/12 = 0.666...

6/10 = 0.6

Since these ratios are not equal, the corresponding sides are not proportional, and therefore the two cuboids are not similar.

Therefore, the answer is "No they are not similar".

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The type of transformation shown is a vertical transformation

In mathematics, a vertical translation refers to a transformation of a function or graph that involves moving it up or down along the y-axis without changing its vertical transformation.

From the attached figure, we can see that the polygons are moved up or down to map them to one another

This means that the transformation is a vertical transformation

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Complete question

Use the image to determine the type of transformation shown.

Image of polygon ABCD and a second polygon A prime B prime C prime D prime below.

Vertical translation

Reflection across the x-axis

180° counterclockwise rotation

Horizontal translation

To find the percentage of students who collected between 49 and 98 kilograms of newspapers, we need to first count the number of students whose collection falls within this range. We can do this by sorting the data and counting the number of values that fall within this range.

Sorting the data, we get:

15, 23, 35, 45, 45, 49, 55, 56, 57, 60, 64, 75, 76, 87, 88, 90, 98, 100, 101, 105, 120, 122

We can see that there are 17 students whose collection falls within the range of 49 to 98 kilograms.

To find the percentage of students, we can divide the number of students whose collection falls within this range by the total number of students and then multiply by 100. The total number of students is the sum of the number of values in the two sets, which is 22 + 22 = 44.

Therefore, the percentage of students who collected between 49 and 98 kilograms of newspapers is:

17/44 * 100% ≈ 38.6%

So approximately 38.6% of the students collected between 49 and 98 kilograms of newspapers.

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

To find the area of the part with cement, we need to find the area of the entire rectangle and then subtract the area of the part with grass.

The area of the rectangle is: length x width = 5 m x 4 m = 20 m²

The area of the part with grass is: length x width = 3 m x 2 m = 6 m²

So the area of the part with cement is:

20 m² - 6 m² = 14 m²

Therefore, the area of the part with cement is 14 square meters.

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a) The demand function is 134.33e^-0.693p

b) At P = $3, we have elasticity is  0.83, at P = $4, we have elasticity is  1.05, at P = $5, we have elasticity is 1.34.

c) We should sell the cheese at a price of $3.84 per pound to maximize monthly revenue.

d) We should sell the cheese at a price of $4.22 per pound to generate the highest revenue within the timeframe of one month.

a) To construct a demand function of the form q = Ae^-bp, we can use the sales figures for the prices $3 and $5 per pound. First, we calculate the values of A and b:

A = q/p = 403/3 ≈ 134.33

b = ln(q/Ap) / p = ln(403/134.33) / (3-5) ≈ 0.693

Using these values, the demand function becomes:

q = 134.33e^-0.693p

b) To find the price elasticity of demand at each of the prices listed, we can use the formula:

E = (dq/dp) * (p/q)

At P = $3, we have:

E = (dq/dp) * (p/q) = (-134.33 * -0.693 * 3) / 403 ≈ 0.83

At P = $4, we have:

E = (dq/dp) * (p/q) = (-134.33 * -0.693 * 4) / 284 ≈ 1.05

At P = $5, we have:

E = (dq/dp) * (p/q) = (-134.33 * -0.693 * 5) / 225 ≈ 1.34

c) To find the price that will maximize monthly revenue, we can use the formula:

p = (1/b) * ln(A/b)

Plugging in the values of A and b that we calculated earlier, we get:

p = (1/0.693) * ln(134.33/0.693) ≈ $3.84

d) If we only have 200 pounds of cheese and it will spoil in one month, we need to sell it at a price that will generate the highest revenue within that timeframe. To do this, we can use the formula:

R = pq

where R is the revenue, p is the price per pound, and q is the quantity sold. We can express q in terms of p using our demand function:

q = 134.33e^-0.693p

Substituting this into the revenue equation, we get:

R = p * 134.33e^-0.693p

To find the price that will maximize revenue, we can take the derivative of R with respect to p and set it equal to zero:

dR/dp = 134.33e^-0.693p - 93.13pe^-0.693p = 0

Solving this equation numerically, we get:

p ≈ $4.22

This price is different from the price calculated in part (c) because we have a limited quantity of cheese that will spoil, so we need to balance the price and quantity sold to maximize revenue within the given timeframe.

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Option A The diameter and circumference of a circle have a proportional relationship is True .

Diameter

The diameter is the length acrοss the circle at its widest pοint, measured frοm center tο center . The radius, a related measurement, is a line that extends frοm the circle's centre tο its edge. The diameter is equivalent tο twice the radius. (A chοrd is a line that crοsses the circle but is nοt at the widest pοint.)

Circumference

The circle's perimeter, οr the distance arοund it, is knοwn as its circumference. Imagine encircling a circle with a string. Imagine taking the string οut and extending it in a straight line. This string's length, if measured, wοuld represent yοur circle's circumference.

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The quotient of the expression (5.04×10^12)÷(6.3×10^9) written in scientific notation is 8 × 10^2.

To divide two numbers written in scientific notation, we can divide their coefficients (the decimal parts) and subtract their exponents. So, we have:

(5.04 × 10^12) ÷ (6.3 × 10^9) = (5.04 ÷ 6.3) × 10^(12-9) = 0.8 × 10^3

Since 0.8 is less than 1, we can write this number in scientific notation by moving the decimal point one place to the right and subtracting 1 from the exponent:

0.8 × 10^3 = 8 × 10^2

Therefore, the quotient of the expression (5.04×10^12)÷(6.3×10^9) written in scientific notation is 8 × 10^2.

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The angle measure of L in the triangle is approximately 75°.

Based on the given table, we can see that the ratio of the opposite leg length to the adjacent leg length for an angle measure of 75° is approximately 3.73. Looking at the triangle in the question, we can see that the side opposite to angle L is the hypotenuse and the adjacent leg is LK.

Therefore, the ratio of the opposite leg length to the adjacent leg length for angle L is equal to the ratio of the hypotenuse length to the length of segment LK.

From the figure, we can see that the length of segment LK is approximately 3.2 units. Therefore, the length of the hypotenuse is approximately 3.73 times the length of segment LK, or:

hypotenuse length ≈ 3.73 × 3.2 ≈ 11.9

Therefore, the angle measure of L is approximately 75°.

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The slant height of the given cone is 16.36 cm.

The length from the base to the peak along the "center" of a lateral face of an object (like a frustum or pyramid) is its slant height.

It is, in other words, the height of the triangle that a lateral face is a part of (Kern and Bland 1948, p.

So, calculate the slant height as follows:

614π = πr√h²+r²

614 = 16.2√h²+16.2²

614 = 262.44√h²

614/262.44 = h

2.33

Height = 2.33 cm

Then, slant height formula:

s=√(r² + h²)

s=√(16.2² + 2.33²)

s=√267.8689

s=16.36 cm

Therefore, the slant height of the given cone is 16.36 cm.

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The customer can choose from 21 different gifts if they want to order only 1 item, 138 different gifts if they want to order 2 items of different kinds, and 280 different gifts if they want to order 3 items of different kinds.

To find the different number of gifts, we use the fundamental counting principle, which states that the total number of ways to choose a sequence of items is equal to the product of the number of choices available for each item.

⇒ If a customer wants to order only 1 item, they can choose from:

10 varieties of "greeting-cards"

4 types of "fruit-baskets"

7 flower arrangements

So, total number of different gifts that can be created is: 10 + 4 + 7 = 21.

⇒ If a customer wants to order 2 items of different kinds, they can choose from:

10 varieties of greeting-cards and 4 types of fruit baskets :

(10 × 4 = 40 choices)

10 varieties of greeting cards and 7 flower arrangements :

(10 × 7 = 70 choices)

4 types of fruit-baskets and 7 flower arrangements :

(4 × 7 = 28 choices)

So, number of different gifts that can be created is: 40 + 70 + 28 = 138.

⇒ If a customer wants to order 3 items of different kinds, they can choose from:

10 varieties of greeting cards, 4 types of fruit baskets, and 7 flower arrangements (10 × 4 × 7 = 280 choices),

So, the total number of different gifts that can be created is : 280.

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Answer:

C

Step-by-step explanation:

first fine the nth term

a+(n-1)d

509+7n+7

516-7n

then equate the ans to the nth term

495=516-7n

7n=516-495

7n= -21

n= -3

The 10 cupcakes will weigh approximately 25.9 ounces when one cubic inch is about 0. 55 ounces, the diameter of the mold is 3 inches and the height of the mold is 2 inches.

Given data:

Diameter of the mold = 3 inches

Radius = 3/2 = 1.5 inches

Height of the mold = 2 inches.

One cubic inch = 0.55 ounces

π = 3. 14

We need to find how many ounces will 10 cupcakes weigh. to find that we need to find the volume of a cone and the weight of one cupcake. we can find  the volume of a cone given by the formula,

[tex]V = (1/3)πr^2h[/tex]

Where:

r = radius

h = height

By Substituting the r and h values into the formula we get:

[tex]V = (1/3)πr^2h[/tex]

[tex]= (1/3) π ((1.5)^2) × (2)[/tex]

[tex]= (1/3) π×(2.25)×(2)[/tex]

= 1.5π

When one cubic inch of the cone is about 0.55 ounces, the weight of one cupcake is approximately

= 1.5π × 0.55

= 0.825π ounces.

The weight of 10 cupcakes is determined by multiplying by the weight of one cupcake, it is given as:

= 10 × 0.825π

= 8.25π ounces

= 8.25 × 3.14

= 25.9 ounces.

Therefore, the 10 cupcakes will weigh approximately 25.9 ounces.

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The TWO statements that represent the relationship between y = 5x and y = log5 x are:

1. They are the exponential and logarithmic form of the same equation.

2. They are inverses of one another.

The two equations are related because they represent the same relationship between x and y, but in different forms. The first equation is an exponential equation, where y is a power of 5 raised to the x power. The second equation is a logarithmic equation, where y is the exponent to which 5 must be raised to get x.

Because the two equations represent the same relationship, they are inverses of one another. If we take the logarithm of both sides of the exponential equation, we get the logarithmic equation. If we raise 5 to both sides of the logarithmic equation, we get the exponential equation. Therefore, the two equations are inverses of one another.

Answer:

2(x - 3) < -24

x - 3 < -12

x < -9

The inequality that represents the given statement is 2(x-3) ≤ -24, where x is the unknown number.

The given statement can be translated into an inequality as "twice the difference of a number (x) and 3 is at most -24". Mathematically, this can be represented as 2(x-3) ≤ -24. Simplifying this inequality, we get 2x - 6 ≤ -24, or 2x ≤ -18, which gives x ≤ -9. Therefore, any number less than or equal to -9 satisfies the given statement.

For example, x = -10 satisfies 2(-10-3) = -26, which is less than or equal to -24. However, any number greater than -9 does not satisfy the given statement. For example, x = -8 gives 2(-8-3) = -22, which is greater than -24. Therefore, the solution set for the given inequality is x ≤ -9.

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The first derivative of f(x) is f'(x) = -12(-2x + 1)2.  The second derivative is f''(x) =48(-2x + 1), and all higher derivatives have the form f^(n)(x) = (-1)n * 6 * n! * (-2x + 1)^(3-n).

To calculate the derivatives of all orders for f(x) = (-2x + 1)3, we first need to find the first derivative:

f(x) = (-2x + 1)³

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

f'(x) = -12(-2x + 1)2

Next, we find the second derivative:

f''(x) = d/dx(-12(-2x + 1)²)

f''(x) = 2(-2)(-12)(-2x + 1)
f''(x) = -12[2(-2x + 1)(-2)]

f''(x)= 48(-2x + 1)

We can continue this process to find the third and fourth derivatives:

f'''(x) = d/dx(96(-2x + 1))

f'''(x) = -384

f''''(x) = d/dx(-384)

f''''(x) = 0
Notice that the fourth derivative is 0, meaning that all higher derivatives will also be 0.

This is because the original function is a polynomial of degree 3, so its fourth derivative will be the derivative of a constant, which is 0.

Therefore, we can conclude that:

f(4)(x) = 0
f(n)(x) = 0 for all n ≥ 4.

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Answer:

1/3

Step-by-step explanation:

There are 3 rectangles, so 1 rectangle is 1 out of 3 rectangles.  So the fraction would be 1/3.

Each rectangle is 1/3 fraction of the complete design.

Fractions are type of numbers which are written in the form p/q, which implies that p parts in a whole of q.

Here p, called the numerator and q, called the denominator, are real numbers.

Given is a design drawn by Mai.

The design consists of three rectangles.

Also, given that each of the rectangle has the same area.

This means that if we find one of the rectangle's area, multiply it by 3 and we will get the whole area.

Or in other words, if we find the whole area, then divide it by 3 to get each of the rectangle's area.

So the required fraction is 1/3.

Hence the fraction is 1/3.

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In ΔPQR, the length of QR is approximately 4.1 feet to the nearest tenth of a foot.

In ΔPQR, given that ∠R=90°, ∠P=26°, and PQ=8.5 feet, you want to find the length of QR to the nearest tenth of a foot.

Step 1: Since ∠R is a right angle (90°), we can use trigonometric ratios to find QR. First, let's find ∠Q. We know that the sum of angles in a triangle is 180°, so ∠Q = 180° - (∠P + ∠R) = 180° - (26° + 90°) = 64°.

Step 2: Now that we have all the angles, we can use the sine formula to find QR. We'll use the sine of ∠P (26°) and the given side PQ (8.5 feet) as our reference. The sine formula is:

QR = (PQ * sin(∠P)) / sin(∠Q)

Step 3: Plug in the known values:

QR = (8.5 * sin(26°)) / sin(64°)

Step 4: Calculate the sine values and the division:

QR = (8.5 * 0.4384) / 0.8988 ≈ 4.1326

Step 5: Round the answer to the nearest tenth of a foot:

QR ≈ 4.1 feet

In ΔPQR, the length of QR is approximately 4.1 feet to the nearest tenth of a foot.

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