Use the image to answer the question. Chem Tech makes paper cones used to filter liquids in lab experiments. To make the paper cones, the company starts with the net of the shape, where the slant height is needed. The company wants the height of the cones to be 6 cm. If they make the diameter 13 cm, what must the slant height be? Use the Pythagorean Theorem and round the answer to the nearest tenth. (1 point) Responses 11.5 cm 11.5 cm 2.5 cm 2.5 cm 8.8 cm 8.8 cm 14.3 cm

Answer 1

To find the slant height of the cone, we can use the Pythagorean theorem. First, we need to determine the radius of the cone since the diameter is given.

1. Calculate the radius:
   Given the diameter is 13 cm, the radius \( r \) is:
   \[ r = \frac{d}{2} = \frac{13 , \text{cm}}{2} = 6.5 , \text{cm} \]

2. Use the Pythagorean theorem:
   In a right triangle formed by the height, the radius, and the slant height, we have: \[ l^2 = h^2 + r^2 \] Where:

   • \( l \) is the slant height
   • \( h = 6 , \text{cm} \) (height of the cone)
   • \( r = 6.5 , \text{cm} \) (radius of the cone)

3. Plug the values into the equation:
   \[ l^2 = 6^2 + (6.5)^2 \] \[ l^2 = 36 + 42.25 \] \[ l^2 = 78.25 \]

4. Calculate the slant height \( l \):
   \[ l = \sqrt{78.25} \approx 8.83 \]

5. Round to the nearest tenth:
   \[ l \approx 8.8 , \text{cm} \]

So, the slant height must be approximately 8.8 cm.