Question

Konery wants to make their waffle cones with a height of 6 in. Their current net design has a diameter of 2.5 in. and a slant height of 6.25 in. Will this work for the desired height? Round the answer to the nearest tenth.(1 point)
Responses

No; the height is too long by 0.7 in. 
No; the height is too long by 0.7 in. 

No; the height is too short by 0.3 in.
No; the height is too short by 0.3 in.

No; the height is too long by 0.1 in.
No; the height is too long by 0.1 in.

No; the height is too long by 0.4 in.

Answer 1

To determine whether the waffle cone design will work for the desired height, we can use the relationship between the slant height, the radius of the base, and the height of a cone.

The slant height (l), height (h), and radius (r) of a cone are related by the Pythagorean theorem:

\[
l^2 = r^2 + h^2
\]

Given:
- Slant height \( l = 6.25 \, \text{in} \)
- Diameter \( d = 2.5 \, \text{in} \) → Radius \( r = \frac{d}{2} = \frac{2.5}{2} = 1.25 \, \text{in} \)
- Desired height \( h = 6 \, \text{in} \)

We need to find out if the height of 6 inches and the slant height of 6.25 inches create a valid cone when the radius is 1.25 inches. We can calculate the height using the equation above:

1. Calculate \( r^2 \):
\[
r^2 = (1.25)^2 = 1.5625
\]

2. Calculate \( l^2 \):
\[
l^2 = (6.25)^2 = 39.0625
\]

3. Substitute into the Pythagorean theorem:
\[
39.0625 = 1.5625 + h^2
\]

4. Solve for \( h^2 \):
\[
h^2 = 39.0625 - 1.5625 = 37.5
\]

5. Now calculate \( h \):
\[
h = \sqrt{37.5} \approx 6.123
\]

Now we compare \( h \approx 6.123 \) inches with the desired height of 6 inches:

The height of the cone is approximately 6.1 inches, which means it is too long by:

\[
6.123 - 6 = 0.123 \text{ inches}
\]

Rounding to the nearest tenth, the cone height is too long by about 0.1 inches.

Thus, the correct response is:
**No; the height is too long by 0.1 in.**