Question

Use the image to answer the question.

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.

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

2.5 cm 

 8.8 cm

14.3 cm

Answer 1

To find the slant height of the cone, we can use the Pythagorean theorem. The height, radius, and slant height form a right triangle, where:

- The height (h) is 6 cm
- The radius (r) is half of the diameter, so for a diameter of 13 cm, the radius is \( r = \frac{13}{2} = 6.5 \) cm
- The slant height (l) is what we need to find

According to the Pythagorean theorem:

\[ l^2 = h^2 + r^2 \]

Substituting the values:

\[ l^2 = 6^2 + 6.5^2 \]

Calculating it step by step:

\[ l^2 = 36 + (6.5 \times 6.5) \]
\[ l^2 = 36 + 42.25 \]
\[ l^2 = 78.25 \]

Now, take the square root to find the slant height:

\[ l = \sqrt{78.25} \approx 8.8 \text{ cm} \]

Therefore, the slant height must be approximately **8.8 cm**.