10) A rectangular building is to be placed on a lot that measures 30 mathrm{~m} by 40 mathrm{~m}. The building must be placed in the lot so that the width of the lawn is the same on all four sides of the building. Local restrictions state that the building cannot occupy any more than 50 % of the property. What are the dimensions of the largest building that can be built on the property?

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Step:1 ：Let&#x27;s denote the width of the lawn as x. Then, the dimensions of the building would be ((30-2x)) by ((40-2x)).Step:2 ：The area of the building would then be ((30-2x)*(40-2x)), and this should be less than or equal to 50% of the total area of the property, which is (30*40/2 = 600) square meters.Step:3 ：We can set up the inequality ((30-2x)*(40-2x) <= 600) and solve for x.Step:4 ：Then, we can substitute x back into ((30-2x)) and ((40-2x)) to find the dimensions of the building.Step:5 ：Final Answer: The dimensions of the largest building that can be built on the property are (boxed{20 mathrm{~m}}) by (boxed{30 mathrm{~m}}).

Answer

Step: 1 ：Let&#x27;s denote the width of the lawn as x. Then, the dimensions of the building would be ((30-2x)) by ((40-2x)).Step: 2 ：The area of the building would then be ((30-2x)*(40-2x)), and this should be less than or equal to 50% of the total area of the property, which is (30*40/2 = 600) square meters.Step: 3 ：We can set up the inequality ((30-2x)*(40-2x) <= 600) and solve for x.Step: 4 ：Then, we can substitute x back into ((30-2x)) and ((40-2x)) to find the dimensions of the building.Step: 5 ：Final Answer: The dimensions of the largest building that can be built on the property are (boxed{20 mathrm{~m}}) by (boxed{30 mathrm{~m}}).

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