To find the slant height of the paper cone, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (in this case, the slant height) is equal to the sum of the squares of the other two sides (the height and the radius of the base of the cone).
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Find the radius:
The diameter of the cone is given as 13 cm, so the radius \( r = \frac{d}{2} = \frac{13}{2} = 6.5 \) cm. -
Use the Pythagorean theorem:
The height \( h \) of the cone is given as 6 cm, and the slant height \( l \) is the hypotenuse. Therefore, using the equation: \[ l^2 = h^2 + r^2 \] Substituting the known values: \[ l^2 = 6^2 + 6.5^2 \] \[ l^2 = 36 + 42.25 \] \[ l^2 = 78.25 \] -
Calculate \( l \):
Now taking the square root to find \( l \): \[ l = \sqrt{78.25} \approx 8.8 \text{ cm} \]
Thus, the slant height needed is approximately 8.8 cm.