To determine if the waffle cone meets the desired height of 6 inches, we can use the relationship between the slant height, the height, and the radius of the cone.
The formula that relates these three dimensions of a right circular cone is derived from the Pythagorean theorem:
\[ s^2 = h^2 + r^2 \]
where:
- \( s \) is the slant height,
- \( h \) is the height,
- \( r \) is the radius (which is half of the diameter).
Given:
- Diameter = 2.5 inches, so the radius \( r = \frac{2.5}{2} = 1.25 \) inches.
- Slant height \( s = 6.25 \) inches.
We need to find the height \( h \) of the cone. Plugging in the known values:
\[ (6.25)^2 = h^2 + (1.25)^2 \]
Calculating \( 6.25^2 \) and \( 1.25^2 \):
\[ 39.0625 = h^2 + 1.5625 \]
Now, isolate \( h^2 \):
\[ h^2 = 39.0625 - 1.5625 \] \[ h^2 = 37.5 \]
Taking the square root of both sides to find \( h \):
\[ h = \sqrt{37.5} \approx 6.124 \]
Now we round this to the nearest tenth:
\[ h \approx 6.1 \]
Now we compare this with the desired height of 6 inches:
\[ 6.1 - 6 = 0.1 \]
This shows that the height is 0.1 inches too long.
Therefore, the correct response is: No, the height is too long by 0.1 in.
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