Question

celeste is designing a water fountain for a city park. she wants one of the spouts to be placed so that it fits within the dimensions shown in the diagram. what is the approximate length of the spout?
a. 16.0 in
b. 25.1 in
c. 33.5 in
d. 29.2 in

Explanation:

Step1: Identify the right triangle

The spout forms the hypotenuse of a right triangle. The base of the triangle is the diagonal of the base rectangle (15 in and 18 in), and the height is 22 in. First, find the length of the base diagonal.
Using the Pythagorean theorem for the base: \( d = \sqrt{15^2 + 18^2} \)
\( 15^2 = 225 \), \( 18^2 = 324 \), so \( d = \sqrt{225 + 324} = \sqrt{549} \approx 23.43 \) in.

Step2: Find the spout length

Now, use the Pythagorean theorem for the triangle with height 22 in and base diagonal ~23.43 in. Let \( L \) be the spout length:
\( L = \sqrt{22^2 + 23.43^2} \)
\( 22^2 = 484 \), \( 23.43^2 \approx 548.96 \)
\( L = \sqrt{484 + 548.96} = \sqrt{1032.96} \approx 32.14 \)? Wait, no, wait. Wait, maybe I misread the dimensions. Wait, maybe the base is 15 and 18, and height 22? Wait, no, maybe the base is a right triangle with legs 15 and 18, and then the spout is the hypotenuse of a triangle with legs (15,18) diagonal and 22? Wait, no, maybe the figure is a rectangular prism, so the spout is the space diagonal. The formula for space diagonal of a rectangular prism is \( \sqrt{l^2 + w^2 + h^2} \), where \( l = 15 \), \( w = 18 \), \( h = 22 \).

Ah, that's the correct approach. So space diagonal \( L = \sqrt{15^2 + 18^2 + 22^2} \)

Step3: Calculate each term

\( 15^2 = 225 \), \( 18^2 = 324 \), \( 22^2 = 484 \)

Step4: Sum the squares

\( 225 + 324 + 484 = 225 + 324 = 549; 549 + 484 = 1033 \)

Step5: Take the square root

\( L = \sqrt{1033} \approx 32.14 \)? Wait, but the options are 16.0, 25.1, 33.5, 29.2. Wait, maybe I misread the dimensions. Wait, maybe the height is 22, and the base is 15 and 18? Wait, let's recalculate:

Wait, 15² + 18² = 225 + 324 = 549. Then 549 + 22² = 549 + 484 = 1033. Square root of 1033 is approx 32.14, but the option C is 33.5, maybe a typo or miscalculation. Wait, maybe the dimensions are 15, 18, and 22? Wait, let's check the options. Wait, maybe I made a mistake. Wait, 15, 18, and 22: let's compute 15² + 18² = 549, 549 + 22² = 1033, sqrt(1033) ≈ 32.14. But option C is 33.5. Alternatively, maybe the height is 22, and the base is 15 and 18, but maybe the problem is a right triangle with legs 18 and 22? No, the figure shows a rectangular prism, so space diagonal. Wait, maybe the numbers are 15, 18, and 22. Let's check the options again. Option C is 33.5, which is close to 32.14, maybe a rounding difference. Alternatively, maybe the dimensions are 15, 18, and 22, and the calculation is:

Wait, 15² = 225, 18² = 324, 22² = 484. Sum: 225 + 324 = 549; 549 + 484 = 1033. Square root of 1033 is approximately 32.14, but the option C is 33.5. Maybe the dimensions are different? Wait, maybe the height is 22, and the base is 15 and 18, but maybe I misread the numbers. Wait, the image shows 15 in, 18 in, and 22 in. Let me check the options again. The options are A.16.0, B.25.1, C.33.5, D.29.2. So 33.5 is the closest to 32.14, maybe due to rounding during calculation. So the approximate length is 33.5 in, which is option C.

Answer:

C. 33.5 in