To determine whether the current design will work for the desired height, we can use the relationship between the slant height, radius, and height of a cone.
Given:
- Diameter = 2.5 inches → Radius \( r = \frac{2.5}{2} = 1.25 \) inches
- Slant height \( l = 6.25 \) inches
- Desired height \( h = 6 \) inches
The relationship between the slant height \( l \), height \( h \), and radius \( r \) of a cone is given by the Pythagorean theorem:
\[ l^2 = r^2 + h^2 \]
Now substituting the known values:
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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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Set up the equation: \[ 39.0625 = 1.5625 + h^2 \]
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Solve for \( h^2 \): \[ h^2 = 39.0625 - 1.5625 = 37.5 \]
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Now, find \( h \): \[ h = \sqrt{37.5} \approx 6.123 \]
Now we can find the difference between the calculated height and the desired height:
\[ \text{Difference} = 6.123 - 6 = 0.123 \text{ inches} \]
Rounding this to the nearest tenth gives 0.1 inches. Since the calculated height is slightly greater than the desired height of 6 inches, we can conclude:
No, the height is too long by 0.1 in.