A drawing shows a circle showing diameter below a sector of a circle showing slant height. A double-headed arrow crossing the center of the circle is labeled as d equals diameter. A dotted double-headed arrow is drawn from the top point of the sector of the circle to a point along the curved edge. The arrow is labeled as lower L equals slant height.

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

14.3 cm
14.3 cm
2.5 cm 
2.5 cm 
 8.8 cm
 8.8 cm
11.5 cm
11.5 cm

Answer 1

To find the slant height of the cone using the Pythagorean theorem, we can use the relationship between the radius, height, and slant height of the cone.

1. Determine the radius:

   • The diameter of the cone is 13 cm, so the radius \( r \) is: \[ r = \frac{d}{2} = \frac{13 , \text{cm}}{2} = 6.5 , \text{cm} \]

2. Given the height:

   • The height \( h \) of the cone is given as 6 cm.

3. Using the Pythagorean Theorem:

   • The slant height \( L \) of the cone, the height \( h \), and the radius \( r \) form a right triangle, where: \[ L^2 = h^2 + r^2 \]
   • Substituting the values: \[ L^2 = 6^2 + (6.5)^2 \] \[ L^2 = 36 + 42.25 \] \[ L^2 = 78.25 \] \[ L = \sqrt{78.25} \approx 8.8 , \text{cm} \]

Thus, the slant height of the cone is approximately 8.8 cm.

The correct response is:

• 8.8 cm