A piece of climbing equipment at a playground is 6 feet high and extends 4 feet horizontally. A piece of climbing equipment at a gym is 10 feet high and extends 6 feet horizontally.

Which statement best compares the slopes of the two pieces of equipment?

A. Because [tex]\(\frac{5}{3} \ \textgreater \ \frac{3}{2}\)[/tex], the slope of the climbing equipment at the gym is greater.
B. Because [tex]\(\frac{5}{3} \ \textless \ \frac{3}{2}\)[/tex], the slope of the climbing equipment at the playground is greater.
C. Because [tex]\(\frac{3}{5} \ \textgreater \ \frac{2}{3}\)[/tex], the slope of the climbing equipment at the gym is greater.
D. Because [tex]\(\frac{3}{5} \ \textless \ \frac{2}{3}\)[/tex], the slope of the climbing equipment at the playground is greater.



Answer :

To compare the slopes of the two pieces of climbing equipment, we need to calculate the slope for each and compare them.

Step-by-Step Solution:

1. Calculate the Slope of the Playground Equipment:

The slope is calculated as the ratio of the height to the horizontal extension.

[tex]\[ \text{slope}_{\text{playground}} = \frac{\text{height}_{\text{playground}}}{\text{horizontal}_{\text{playground}}} = \frac{6 \text{ feet}}{4 \text{ feet}} = \frac{6}{4} = 1.5 \][/tex]

2. Calculate the Slope of the Gym Equipment:

Similarly, calculate the slope for the gym equipment.

[tex]\[ \text{slope}_{\text{gym}} = \frac{\text{height}_{\text{gym}}}{\text{horizontal}_{\text{gym}}} = \frac{10 \text{ feet}}{6 \text{ feet}} = \frac{10}{6} = \frac{5}{3} \approx 1.67 \][/tex]

3. Compare the Slopes:

Now, we compare the two calculated slopes:

[tex]\[ \text{slope}_{\text{playground}} = 1.5 \][/tex]
[tex]\[ \text{slope}_{\text{gym}} = \frac{5}{3} \approx 1.67 \][/tex]

Clearly, [tex]\( 1.67 > 1.5 \)[/tex]. Thus, the slope of the climbing equipment at the gym is greater than the slope of the climbing equipment at the playground.

4. Verify the Correct Statement:

Based on the comparison, the correct statement is:

[tex]\[ \text{Because } \frac{5}{3} > \frac{3}{2}, \text{ the slope of the climbing equipment at the gym is greater.} \][/tex]

Therefore, the correct answer is:

Because [tex]\(\frac{5}{3} > \frac{3}{2}\)[/tex], the slope of the climbing equipment at the gym is greater.