A museum is being built on a lot that measures 80 m by 60 m. A lawn of uniform width that is 40% of the area of the museum must surround the museum on all sides. What dimensions must the museum have?

Question

this is for my homework and I extremely need help for it.

Answer 1

If the width of the lawn is w and the dimensions of the rectangular museum are a and b, the area of the lawn is 60×80=4800-ab=0.4ab square metres.

So 4800=1.4ab. The dimensions of the ground area are a+2w and b+2w because the width w of the lawn is uniform.

Therefore:

a+2w=80, a=80-2w; b+2w=60, b=60-2w;

4800=1.4(80-2w)(60-2w)=1.4(4800-280w+4w²).

4800=6720-392w+5.6w²,

5.6w²-392w+1920=0,

w²-70w=-2400/7,

w²-70w+1225=1225-2400/7=6175/7,

(w-35)²=6175/7,

w=35±√(6175/7)=64.7m, 5.3m (approx).

We can reject 64.7m because this value would lead to negative values for a and b; therefore, a=80-10.6=69.4m, b=49.4m.

CHECK

Area of museum (footprint)=3428.36m² approx.

Area of lawn=4800-3428.36=1371.64m² approx.

1371.64/3428.36=0.40 approx (40%).