The formula for the lateral surface area of a cone is A=πr√(h^2+r^2). (A)Solve for r and simplify the result, if possible. (B) If the lateral surface area of a cone is 10 square feet and the height is 16 inches, what is the radius of the cone in inches? Round your answer to the nearest hundredth of an inch. (C) What is this radius in feet? Please help!!!! :(
A) Sure thing! Let's solve the formula for r.
A = πr√(h^2 + r^2)
First, let's square both sides:
A^2 = (πr√(h^2 + r^2))^2
A^2 = π^2r^2(h^2 + r^2)
Now, let's multiply both sides by (h^2 + r^2):
A^2(h^2 + r^2) = π^2r^2(h^2 + r^2)
Expanding the equation:
A^2h^2 + A^2r^2 = π^2r^2h^2 + π^2r^4
Let's rearrange the terms:
A^2h^2 - π^2r^2h^2 = π^2r^4 - A^2r^2
(h^2(A^2 - π^2r^2)) = r^2(π^2r^2 - A^2)
Now, let's isolate r:
r^2 = (h^2(A^2 - π^2r^2)) / (π^2r^2 - A^2)
r^2(π^2r^2 - A^2) = h^2(A^2 - π^2r^2)
π^2r^4 - A^2r^2 = A^2h^2 - π^2h^2r^2
π^2r^4 - π^2h^2r^2 + A^2r^2 = A^2h^2
Now, let's factor out r^2:
r^2(π^2r^2 - π^2h^2 + A^2) = A^2h^2
Finally, divide both sides by (π^2r^2 - π^2h^2 + A^2) to solve for r:
r^2 = (A^2h^2) / (π^2r^2 - π^2h^2 + A^2)
r = √[(A^2h^2) / (π^2r^2 - π^2h^2 + A^2)]
B) Let's substitute the given values into the formula:
A = 10 square feet = 10 ft^2
h = 16 inches = 16/12 = 1.33 feet
Now, we can calculate the radius using the formula:
r = √[(A^2h^2) / (π^2r^2 - π^2h^2 + A^2)]
r = √[(10^2 * 1.33^2) / (π^2r^2 - π^2 * 1.33^2 + 10^2)]
Simplifying and solving gives us the radius in feet.
C) To convert the radius from feet to inches, we can multiply the radius by 12 since there are 12 inches in a foot.
A) To solve for r in the formula A=πr√(h^2+r^2), we need to isolate r.
Start by dividing both sides of the equation by π:
A/π = r√(h^2+r^2)
Next, square both sides of the equation to eliminate the square root:
(A/π)^2 = (r√(h^2+r^2))^2
(A/π)^2 = r^2(h^2+r^2)
Expand the equation:
(A^2/π^2) = r^2(h^2+r^2)
Multiply both sides of the equation by π^2:
π^2(A^2/π^2) = π^2(r^2(h^2+r^2))
A^2 = π^2r^2(h^2+r^2)
Rearrange the equation to solve for r:
A^2 = (h^2+r^2)π^2r^2
A^2 = (h^2+r^2)(πr)^2
Take the square root of both sides to solve for r:
√(A^2) = √(h^2+r^2)(πr)^2
A = rπ(h^2+r^2)
Finally, divide both sides of the equation by π(h^2+r^2):
A/π(h^2+r^2) = r
B) Let's substitute the given values into the formula A=πr√(h^2+r^2) and solve for r:
A = 10 square feet
h = 16 inches
A = πr√(h^2+r^2)
10 = πr√(16^2+r^2)
Square both sides of the equation to eliminate the square root:
100 = π^2r^2(16^2+r^2)
Divide both sides of the equation by π^2(16^2+r^2):
100 / π^2(16^2+r^2) = r^2
Take the square root of both sides to solve for r:
r = √(100 / π^2(16^2+r^2))
r = 4.80 inches (rounded to the nearest hundredth of an inch)
C) To convert the radius from inches to feet, divide by 12 (since there are 12 inches in a foot):
4.80 inches / 12 = 0.40 feet
Therefore, the radius of the cone is 0.40 feet.
(A) To solve for r in the formula A=πr√(h^2+r^2), we need to isolate r on one side of the equation.
Start by dividing both sides of the equation by π:
A/π = r√(h^2+r^2)
Next, square both sides of the equation to eliminate the square root:
(A/π)^2 = (r√(h^2+r^2))^2
Simplifying further:
(A/π)^2 = r^2(h^2+r^2)
Distribute r^2 on the right side of the equation:
(A/π)^2 = r^2h^2 + r^4
Move the r^2h^2 term to the left side of the equation:
(A/π)^2 - r^2h^2 = r^4
Combine like terms:
r^4 - r^2h^2 = (A/π)^2
Now, we have a quadratic equation in terms of r^2. We can solve for r^2 and then find r by taking the square root of r^2.
(B) Given that the lateral surface area (A) is 10 square feet and the height (h) is 16 inches, we can now substitute these values into the equation and solve for r.
10/π = r^2(16^2) + r^4
Simplify the equation further:
10/π = 256r^2 + r^4
Now, we have a quadratic equation. Rearrange the terms to bring the equation to standard quadratic form:
r^4 + 256r^2 - (10/π) = 0
To solve this equation, you can use a numerical solver or approximation methods like factoring or the quadratic formula to find the values of r. Once you find the value(s) of r, you can round the answer to the nearest hundredth of an inch, as requested.
(C) Once you find the radius (r) in inches, you can convert it to feet by dividing it by 12 since there are 12 inches in one foot. Divide the radius in inches by 12 to get the radius in feet.