To determine the height [tex]\( h \)[/tex] of the right cone, we'll use the Pythagorean Theorem. Here are the steps:
1. Understand the Given Data:
- The slant height [tex]\( l \)[/tex] of the cone is [tex]\( 17 \)[/tex] feet.
- The diameter of the base of the cone is [tex]\( 30 \)[/tex] feet.
2. Determine the Radius of the Base:
- The radius [tex]\( r \)[/tex] can be found by dividing the diameter by 2.
[tex]\[
r = \frac{30}{2} = 15 \text{ feet}
\][/tex]
3. Apply the Pythagorean Theorem:
- In a right cone, the slant height [tex]\( l \)[/tex], the radius [tex]\( r \)[/tex], and the height [tex]\( h \)[/tex] form a right triangle.
- The Pythagorean Theorem states that:
[tex]\[
l^2 = r^2 + h^2
\][/tex]
- Substitute the known values:
[tex]\[
17^2 = 15^2 + h^2
\][/tex]
4. Simplify and Solve for [tex]\( h \)[/tex]:
- Calculate [tex]\( 17^2 \)[/tex] and [tex]\( 15^2 \)[/tex]:
[tex]\[
17^2 = 289
\][/tex]
[tex]\[
15^2 = 225
\][/tex]
- Substitute these values back into the equation:
[tex]\[
289 = 225 + h^2
\][/tex]
- Isolate [tex]\( h^2 \)[/tex] by subtracting 225 from both sides:
[tex]\[
h^2 = 289 - 225
\][/tex]
- Simplify the right side:
[tex]\[
h^2 = 64
\][/tex]
- Take the square root of both sides to solve for [tex]\( h \)[/tex]:
[tex]\[
h = \sqrt{64}
\][/tex]
- Therefore,
[tex]\[
h = 8 \text{ feet}
\][/tex]
Thus, the height [tex]\( h \)[/tex] of the cone is [tex]\( 8 \)[/tex] feet. The correct answer is:
A. [tex]\( 8 \text{ ft} \)[/tex]
