A conical paper cup is to have a height of 3 inches. Find the radius r of the cone that will result in a surface area of 6ðin^2.
To find the radius (r) of a conical paper cup with a given height (h) and surface area (A), we can use the following formula:
A = π * r * (r + √(r^2 + h^2))
In this case, we have the height (h) as 3 inches and the surface area (A) as 6π in^2. We can substitute these values into the formula to find the radius (r).
6π = π * r * (r + √(r^2 + 3^2))
Simplifying the equation:
6 = r * (r + √(r^2 + 9))
Now, let's solve this quadratic equation to find the value of r.
Expanding and rearranging the equation:
6r = r^2 + r√(r^2 + 9)
Subtracting 6r from both sides:
r^2 - 6r + r√(r^2 + 9) = 0
Factoring out r:
r(r - 6 + √(r^2 + 9)) = 0
From this equation, we can see that either r = 0 or r - 6 + √(r^2 + 9) = 0. Since r cannot be zero in this context, we will focus on the second equation:
r - 6 + √(r^2 + 9) = 0
Adding 6 to both sides:
r + √(r^2 + 9) = 6
Subtracting r from both sides:
√(r^2 + 9) = 6 - r
Squaring both sides to eliminate the square root:
r^2 + 9 = (6 - r)^2
Expanding and simplifying:
r^2 + 9 = 36 - 12r + r^2
Moving all terms to one side:
0 = 36 - 12r - r^2 + r^2 - 9
Rearranging the equation:
0 = 27 - 12r
Moving 27 to the other side:
12r = 27
Dividing both sides by 12:
r = 27/12
Simplifying the fraction:
r = 9/4
Therefore, the radius (r) of the conical paper cup that will result in a surface area of 6π in^2 is 9/4 inches.