To determine if Koney's current net design will work for the desired height, we can use the geometry of a cone.
We have:
- Diameter of the base (d) = 2.5 in
- Radius (r) = d/2 = 2.5 in / 2 = 1.25 in
- Slant height (l) = 6.25 in
- Desired height (h) = 6 in
To find the actual height of the cone from the given slant height and radius, we can use the Pythagorean theorem. The relationship is:
\[ l^2 = h^2 + r^2 \]
Now, we can plug in the known values:
\[ 6.25^2 = h^2 + 1.25^2 \] \[ 39.0625 = h^2 + 1.5625 \] Subtract \(1.5625\) from both sides:
\[ 39.0625 - 1.5625 = h^2 \] \[ 37.5 = h^2 \] Now take the square root of both sides:
\[ h = \sqrt{37.5} \approx 6.123 \]
Now, we can compare this calculated height to the desired height of 6 inches:
\[ 6.123 - 6 = 0.123 \text{ in} \]
Since we are rounding to the nearest tenth, this shows that the height is too long by approximately 0.1 inches.
Thus, the answer is:
No, the height is too long by 0.1 in.