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.

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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