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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.

Respuesta :

JcAlmighty
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

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