Question
Find the height, h, of the cylinder. The radius, r, is 5 cm and the diagonal, d, is 15 cm. Round the answer to the nearest tenth.
(1 point)
Responses
11.2 cm
11.2 cm
15.8 cm
15.8 cm
18.0 cm
18.0 cm
14.1 cm
Answers
GPT-4o mini
To find the height \( h \) of the cylinder given the radius \( r \) and the diagonal \( d \), we can use the relationship derived from the Pythagorean theorem.
The diagonal \( d \) of the cylinder forms a right triangle with the height \( h \) of the cylinder and the diameter of the base of the cylinder. The diameter \( D \) is twice the radius, so:
\[
D = 2r = 2 \times 5 \, \text{cm} = 10 \, \text{cm}
\]
The formula relating the height, radius, and diagonal for the cylinder is:
\[
d^2 = h^2 + D^2
\]
Substituting in the known values:
\[
15^2 = h^2 + 10^2
\]
Calculating the squares:
\[
225 = h^2 + 100
\]
Now, we can isolate \( h^2 \):
\[
h^2 = 225 - 100
\]
\[
h^2 = 125
\]
Taking the square root of both sides gives us:
\[
h = \sqrt{125} \approx 11.1803
\]
Rounding to the nearest tenth, we find:
\[
h \approx 11.2 \, \text{cm}
\]
Thus, the height of the cylinder is:
**11.2 cm**
The diagonal \( d \) of the cylinder forms a right triangle with the height \( h \) of the cylinder and the diameter of the base of the cylinder. The diameter \( D \) is twice the radius, so:
\[
D = 2r = 2 \times 5 \, \text{cm} = 10 \, \text{cm}
\]
The formula relating the height, radius, and diagonal for the cylinder is:
\[
d^2 = h^2 + D^2
\]
Substituting in the known values:
\[
15^2 = h^2 + 10^2
\]
Calculating the squares:
\[
225 = h^2 + 100
\]
Now, we can isolate \( h^2 \):
\[
h^2 = 225 - 100
\]
\[
h^2 = 125
\]
Taking the square root of both sides gives us:
\[
h = \sqrt{125} \approx 11.1803
\]
Rounding to the nearest tenth, we find:
\[
h \approx 11.2 \, \text{cm}
\]
Thus, the height of the cylinder is:
**11.2 cm**