Math CentralQuandaries & Queries


Question from jessica, a student:

A Landscaper, who just completed a rectangular flower garden measuring 6 feet by 10 feet, orders 1 cubic yard of premixed cement, all of which is to be used to create a border of uniform width around the garden. if the border is to have a depth of 3 inches, how wide will the border be?
( 1 cubic yard=27 cubic feet)

Hi Jessica,

I am going to choose feet as the dimensions to use. Hence the depth of the border is $3$ inches which is $\frac{3}{12} = \frac14$ feet. The volume of the concrete (27 cubic feet) is the surface area times the depth so

\[27 = \mbox{ surface area } \times \frac14.\]

And hence the surface area of the concrete is $4 \times 27$ square feet.

If the border is $d$ feet wide then the flower garden plus the border is $6 + 2d$ feet by $10 + 2d$ feet. The surface area of the concrete is then $(6 + 2d) \times (10 + 2d)$ square feet minus the surface area of the garden. Set this equal to $4 \times 27$ and solve for $d.$


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