An illustration shows a cylinder resting on its bottom base. The top base is visible. The part of the bottom base that is not visible is indicated by a dashed line. Parts of the cylinder are labeled. A large black point is shown in the center of the top face of the cylinder. A solid line extends from that point to the edge of the cylinder and is labeled r indicating the radius. The vertical line on the right side of the cylinder is labeled h. On the interior of the cylinder, a dotted line extends diagonally from top right to bottom left from where the radius line meets the edge down to the opposite side of the cylinder where the other face meets the edge. This line is labeled d.

Question

An illustration shows a cylinder resting on its bottom base. The top base is visible. The part of the bottom base that is not visible is indicated by a dashed line. Parts of the cylinder are labeled. A large black point is shown in the center of the top face of the cylinder. A solid line extends from that point to the edge of the cylinder and is labeled r indicating the radius. The vertical line on the right side of the cylinder is labeled h. On the interior of the cylinder, a dotted line extends diagonally from top right to bottom left from where the radius line meets the edge down to the opposite side of the cylinder where the other face meets the edge. This line is labeled d.
Find the height,  h
 , of the cylinder. The radius,  r
 , is 5 cm and the diagonal,  d
 , is 15 cm. Round the answer to the nearest tenth.

(1 point)
Responses

18.0 cm
18.0 cm

11.2 cm
11.2 cm

14.1 cm
14.1 cm

15.8 cm

GPT-4o mini · Answer

To find the height $ h $ of the cylinder, we can use the Pythagorean theorem since we are dealing with a right triangle formed by the radius $ r $, the height $ h $, and the diagonal $ d $. The relationship can be expressed as follows: $$ d^2 = r^2 + h^2 $$ We know: - $ r = 5 $ cm - $ d = 15 $ cm Substituting these values into the equation: $$ 15^2 = 5^2 + h^2 $$ Calculating the squares: $$ 225 = 25 + h^2 $$ Now, isolate $ h^2 $: $$ h^2 = 225 - 25 $$ $$ h^2 = 200 $$ Taking the square root of both sides to find $ h $: $$ h = \sqrt{200} $$ $$ h = \sqrt{100 \times 2} = 10\sqrt{2} $$ $$ h \approx 10 \times 1.414 = 14.14 \text{ cm} $$ Rounding to the nearest tenth, we get: $$ h \approx 14.1 \text{ cm} $$ Thus, the height $ h $ of the cylinder is: **14.1 cm**

cripface · Answer

wrong

GPT-4o mini · Answer

Let’s go through the calculation again, step by step, to find the height $ h $ of the cylinder using the given radius $ r $ and diagonal $ d $.

Step 1: Formula Setup

We use the Pythagorean theorem, where the diagonal $ d $ is the hypotenuse and the height $ h $ and radius $ r $ are the two legs of the triangle:

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

Step 2: Substitute Given Values

Given:

$ r = 5 $ cm
$ d = 15 $ cm

Substituting these values into the equation:

\[ 15^2 = 5^2 + h^2 \]

Step 3: Calculate the Squares

Calculating the squares:

\[ 225 = 25 + h^2 \]

Step 4: Isolating $ h^2 $

Now isolate $ h^2 $:

\[ h^2 = 225 - 25 \] \[ h^2 = 200 \]

Step 5: Take the Square Root

Now, take the square root to find $ h $:

\[ h = \sqrt{200} \]

Step 6: Simplifying the Square Root

We can simplify:

\[ h = \sqrt{100 \times 2} = 10\sqrt{2} \] \[ \sqrt{2} \approx 1.414 \] \[ h \approx 10 \times 1.414 = 14.14 \text{ cm} \]

Step 7: Round the Answer

Rounding to the nearest tenth:

\[ h \approx 14.1 \text{ cm} \]

If you believe this is not the correct answer, please let me know what method you would like me to use or if there was any misunderstanding in the problem setup. Based on typical geometric calculations, the height calculated should be correct for the given parameters.