The house to build has 91 feet of frontage on a lake and is 158 feet deep. The house can be no closer than 10 feet to the lot line. Write an inequality and solve to see how long the front of the house facing the lake may be.

max hous wide next tu lake=91-20=71
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1. If the frontage refers to the length (91 feet) of the front side of the house facing the lake, and if the land on which the house is to be built is 158 feet from the lake (and this land may be sloping gradually down to the lake 158 feet below the level of the house and lot), and if the house has to be no closer than 10 feet from the lot line and the lot line is 158 feet from the lake, the front of the house must be greater than or equal to 158+10=168 feet. The length of the front of the house is not involved.
2. If the frontage refers to 91 feet of land between the lot line and the lake, then the house must be built no closer than 91+10 feet = 101 feet from the lake. If the lake is 158 feet deep, this depth has no relevance to the answer, and the house must face the lake and be built > 101 feet from the edge of the lake.
3. To use all the figures meaningfully, the house is > 10 feet from the lot line, the lot line is 91 feet (frontage) from the edge of the lake, which is 158 feet below the level of the lot line. The house is > 101 feet from the point where the ground slopes down to the lake from the lot line. We need to add the length of the slope to the distance between the house and the lot line, but we only know the depth of the slope; we don't know the angle. If this slope length is L then 158/L=sinx where x is the slope angle. If x is not zero (there is a slope) then L=158cosecx. Since cosec has a minimum value of 1, making the minimum value of L 158 feet. So the total distance from the house to the lake is > L+101, or > 259 feet.
4. In this interpretation, the land slopes away from the house all the way down to the lake so that the depth of slope is 158 feet. The frontage is 91 feet measured from the lot line to the lake's edge along the slope. The land from the house to the lot line is at least 10 feet of slope. This time the angle of slope can be calculated: sinx=158/(10+91)>1, so this interpretation is invalid because the angle of slope is not possible (sinx cannot exceed 1).

Conclusion

The best answer using all the figures given is (3), making the distance from the house to the lake > 259 feet. This would be the length of frontage from the house to the lake, not the length of the front the house facing the lake, which is unaffected by its distance from the lake (91 feet being the length of the front face of the house---see (1)).

by Top Rated User (716k points)

Length = 91 - 2*10 = 71 ft

by Level 8 User (30.1k points)