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:
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Calculate \( r^2 \): \[ r^2 = (1.25)^2 = 1.5625 \]
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Calculate \( l^2 \): \[ l^2 = (6.25)^2 = 39.0625 \]
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Substitute into the Pythagorean theorem: \[ 39.0625 = 1.5625 + h^2 \]
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Solve for \( h^2 \): \[ h^2 = 39.0625 - 1.5625 = 37.5 \]
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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.