To determine if the current design will work for the desired height of the waffle cone, we can use the relationship between the cone's height (h), the radius (r), and the slant height (l) using the Pythagorean theorem in a right triangle.
Given:
- Diameter of the cone = 2.5 in, so the radius \( r = \frac{2.5}{2} = 1.25 \) in.
- Slant height \( l = 6.25 \) in.
- Desired height \( h = 6 \) in.
According to the Pythagorean theorem: \[ l^2 = h^2 + r^2 \]
We can find the height \( h \) using the slant height \( l \) and the radius \( r \):
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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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Rearranging the Pythagorean theorem to find height \( h \): \[ h^2 = l^2 - r^2 \] \[ h^2 = 39.0625 - 1.5625 = 37.5 \] \[ h = \sqrt{37.5} \approx 6.123 \]
The calculated height of the cone is approximately \( 6.1 \) in.
Now to find out if this meets the desired height of \( 6 \) in: \[ 6.1 - 6 = 0.1 ; \text{in} \]
Since the actual height of the cone is \( 6.1 \) in, which is \( 0.1 \) in too long, the correct response is: No; the height is too long by 0.1 in.