The circumference of the top rim of the cone-shaped paper cup is 7.17 inches. Find the least amount of paper that can form the cone-shaped cup. (Round your answer to two decimal places.
*I realized that I did not put the height (3.9in) in the question above.
I am not sure how I got this right on the quiz, but I did! :)
This is the steps I wrote:
C=7.17
r=3.58
pi*3.58*/pi*3.9
=3.58*3.9
=13.98 in^2
Please, could anyone show me the correct formula steps on how to get to the correct work shown. I am still not entirely sure how I even came up with this.
lateral area of a cone of radius r and slant height s is
A = πrs = πr√(r^2+h^2)
since C = 2πr,
A = π(C/2π)√((C/2π)^2+h^2) = C/4π √(C^2 + (2πh)^2)
The formulas you used were
r = C/2
A = rh
I cannot see how it was even close
To find the least amount of paper that can form the cone-shaped cup, we need to find the slant height of the cone.
The formula for the circumference of a circle is:
C = 2πr
Since the top rim of the cone is a circle, the circumference of the top rim can be written as:
7.17 = 2πr
To find the radius (r), we divide both sides of the equation by 2π:
r = 7.17 / (2π)
Now, we need to find the slant height (l) of the cone using the following formula:
l = √(r^2 + h^2)
Since the cup is a cone, the height (h) of the cone will be equal to the slant height (l), so we can rewrite the formula as:
l = √(r^2 + l^2)
We can substitute the value of r we found earlier into this equation:
l = √((7.17/(2π))^2 + l^2)
To find the value of l, we can rearrange the equation and solve for l:
l^2 = [((7.17/(2π))^2] / [1 - 1/π^2]
Using a calculator, we can find that l ≈ 4.25 inches.
Now, to find the least amount of paper that can form the cone-shaped cup, we need to calculate the surface area of the cone.
The formula for the surface area of a cone is:
A = πr(r + l)
Substituting the values of r and l we found into this equation:
A = π * (7.17/(2π)) * (7.17/(2π) + 4.25)
Using a calculator, we can find that the surface area (A) is approximately 54.43 square inches.
Therefore, the least amount of paper that can form the cone-shaped cup is approximately 54.43 square inches.
To find the least amount of paper that can form the cone-shaped cup, we need to calculate the surface area of the cone.
The formula for the surface area of a cone is given by:
SA = πr(r + √(r^2 + h^2))
where SA is the surface area, r is the radius of the top rim, and h is the height of the cone.
In this case, we are given the circumference of the top rim, which is equal to 2πr. So, we can rearrange the formula to solve for r:
2πr = 7.17
To find r, we divide both sides of the equation by 2π:
r = 7.17 / (2π)
Next, we need to find the height of the cone. Since we are not given the height directly, we will use the Pythagorean theorem to find it. The height can be considered as the slant height or the hypotenuse of a right triangle, where the radius is the base and the height is the perpendicular side.
We can use the formula:
h = √(l^2 - r^2)
where l is the slant height. The slant height can be found using the circumference and the radius:
l = circumference / (2π) = 7.17 / (2π)
Now, we can substitute the values of r and l into the formula to find h:
h = √(l^2 - r^2) = √((7.17 / (2π))^2 - (7.17 / (2π))^2)
After finding the values of r and h, we can substitute them back into the formula of the surface area to calculate the least amount of paper required:
SA = πr(r + √(r^2 + h^2))
Finally, round the answer to two decimal places to find the least amount of paper required to form the cone-shaped cup.