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.
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.
