A rectangular garden has width \(x\) metres and length \((x + 7)\) metres. A fence is built all the way around the garden. (a) Write an expression, in terms...
Question 1 Report
A rectangular garden has width \(x\) metres and length \((x + 7)\) metres. A fence is built all the way around the garden.
(a) Write an expression, in terms of \(x\), for the perimeter of the garden. Give your answer in its simplest form. [2]
(b) The fence costs $12 for each metre. Write an expression, in terms of \(x\), for the total cost of the fence. [1]
(c) The total cost of the fence is $840. Find the value of \(x\). [2]
Answer Details
A fence around a garden follows its perimeter, so this question links perimeter, a cost per metre, and a linear equation.
(a) The width is \(x\) m and the length is \((x + 7)\) m, and a rectangle has two of each:
\[ P = 2x + 2(x + 7) \] [M1]
\[ P = 2x + 2x + 14 = 4x + 14 \ \text{m} \] [A1]
(b) Each metre of fence costs \(\$12\), so the total cost is \(12\) multiplied by the number of metres:
\[ \text{Cost} = 12(4x + 14) = 48x + 168 \] [B1]
Either form is acceptable. The whole perimeter must be multiplied, not just the \(4x\), since every metre is charged the same rate.
(c) Set the cost expression equal to \(\$840\):
\[ 48x + 168 = 840 \quad \Rightarrow \quad 48x = 672 \] [M1]
\[ x = \frac{672}{48} = 14 \] [A1]
Check the whole chain: with \(x = 14\) the garden is \(14\) m by \(21\) m, so the perimeter is \(2(14) + 2(21) = 70\) m, and \(70 \times 12 = \$840\), as given.
A frequent misreading is to use the area instead of the perimeter. A fence encloses the boundary, so it is the distance around that is paid for, not the ground covered.
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