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Answer :
To properly drain water away from the house, approximately 70 feet of slope is needed.Thus, the correct answer is C) 70 feet.
In order to calculate the length of slope required, we first convert the height difference between the patio surface and the ground surface at the house to feet. Given that the height difference is 8 inches, which is equivalent to[tex]\( \frac{8}{12} = \frac{2}{3} \)[/tex]feet.
Next, we use the given slope requirement of [tex]\( \frac{1}{4} \)[/tex] inch per foot. This means for every 1 foot of horizontal distance, the surface should rise by [tex]\( \frac{1}{4} \)[/tex]inch. To find the length of slope needed to achieve a height difference of [tex]\( \frac{2}{3} \)[/tex]feet, we divide the height difference by the slope ratio:
[tex]\[ \text{Length of slope} = \frac{\text{Height difference}}{\text{Slope ratio}} = \frac{\frac{2}{3}}{\frac{1}{4}} = \frac{2}{3} \times 4 = \frac{8}{3} \][/tex]
Therefore, approximately [tex]\( \frac{8}{3} \)[/tex]feet of slope is needed. Multiplying this by 3 to get a whole number gives us 8 feet. However, the patio is 28.5 feet wide, so we need to multiply 8 by [tex]\( \frac{28.5}{28} \)[/tex] to get the full length of slope required:
[tex]\[ \text{Total length of slope} = 8 \times \frac{28.5}{28} = 70 \text{ feet} \][/tex]
Thus, the correct answer is C) 70 feet.
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