To determine if the current net design of the waffle cone will work for a desired height of 6 inches, we need to check the relationship between the slant height, the height, and the radius of the 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:
- Desired height \( h = 6 \) inches
- Diameter \( d = 2.5 \) inches, so the radius \( r = \frac{d}{2} = \frac{2.5}{2} = 1.25 \) inches
- Current slant height \( l = 6.25 \) inches
Now we can substitute these values into the Pythagorean theorem equation to see if they are consistent:
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Calculate \( r^2 \): \[ r^2 = (1.25)^2 = 1.5625 \]
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Calculate \( h^2 \): \[ h^2 = (6)^2 = 36 \]
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Now add \( r^2 \) and \( h^2 \): \[ r^2 + h^2 = 1.5625 + 36 = 37.5625 \]
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Calculate \( l^2 \): \[ l^2 = (6.25)^2 = 39.0625 \]
Now we compare \( l^2 \) and \( r^2 + h^2 \):
- \( r^2 + h^2 = 37.5625 \)
- \( l^2 = 39.0625 \)
Since \( l^2 \) (39.0625) is greater than \( r^2 + h^2 \) (37.5625), the relationship holds true, meaning that a cone with these dimensions can have a height of 6 inches and a slant height of 6.25 inches.
Therefore, the current net design will work for the desired height of 6 inches.
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