Trina has $1000 to purchase an open-top cylindrical dog pen in her backyard. She wants the height of the pen to be 5 feet. If the pen costs $1 per square foot, what is the biggest pen (in terms of the radius) that she can afford? Round your answer to the nearest foot.

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Mathematics

Guest

31 December, 13:34

Trina has $1000 to purchase an open-top cylindrical dog pen in her backyard. She wants the height of the pen to be 5 feet. If the pen costs $1 per square foot, what is the biggest pen (in terms of the radius) that she can afford? Round your answer to the nearest foot.

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Irene · Answer 1

The answer is 13 ft.

The surface area of the cylinder (A) is the sum of the area of its two bases (A1) and lateral surface area (A2) : A = 2 * A1 + A2

The surface area of the open-top cylinder (A) is the sum of the area of its one base (A1) and lateral surface area (A2) : A = A1 + A2

The base of the cylinder is a circle and its area is: A1 = r²*π

The lateral surface area of cylinder is: A2 = 2*r*h*π

So, the surface area of the open-top cylindrical dog pen is:

A = r²*π + 2*r*h*π

She wants the height of the pen to be 5 feet: h = 5 ft.

If the pen costs $1 per square foot, for $1000 she can by 1000 square foot. Thus: A = 1000 ft².

r²*π + 2*r*h*π = 1000

π (r² + 2*r*h) = 1000

3.14 * (r² + 2*r*5) = 1000

r² + 2*r*5 = 1000/3.14

r² + 10r = 318

r² + 10r - 318 = 0

Using the formula for quadratic function:

r = (-b+/-√ (b² - 4ac) / 2a

= (-10 + / - √ (10² - 4 * 1 * (-318))) / (2*1)

= (-10 + / - √ (100 + 1272) / 2

= (-10 + / - √1372) / 2

= (-10 + / - 37) / 2

r = (-10 - 37) / 2 = - 47/2 = - 23.5 (it's negative, so cannot be radius

or

r = (-10 + 37) / 2 = 13.5

So, we need to round it correctly.

If r = 13, then:

A = r²*π + 2*r*h*π = 13²*3.14 + 2*13*5*3.14 = 530.66 + 408.2 = 938.86 ft²

So, A < 1000

If r = 14, then:

A = r²*π + 2*r*h*π = 14²*3.14 + 2*14*5*3.14 = 615.44 + 439.6 = 1055.04 ft²

So, A > 1000

We need A < 1000, so r = 13 ft