A rectangular plot consists of a rectangular pond surrounded by a path. The length and breadth of the plot are $x$ metres and $y$ metres respectively. The path is 1.5 metres wide at the ends of the pond and 1 metre wide along the other sides as shown. The total area of the pond and path together is 150 square metres. (a) Show that the area of the pond, $A$ square metres, is given by $$A(x) = 156 - 2x - \frac{450}{x}.$$ (b) Determine the maximum area of the pond.

Scottish Highers Maths Differentiation: A rectangular plot consists of a

A rectangular plot consists of a rectangular pond surrounded by a path - Scottish Highers Maths - Question 8 - 2022

Question 8

A rectangular plot consists of a rectangular pond surrounded by a path. The length and breadth of the plot are $x$ metres and $y$ metres respectively. The path is 1... show full transcript

Worked Solution & Example Answer:A rectangular plot consists of a rectangular pond surrounded by a path - Scottish Highers Maths - Question 8 - 2022

Step 1

(a) Show that the area of the pond, $A$ square metres, is given by

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Answer

To find the area of the pond, we start by determining the dimensions based on the path width.

The total area of the plot (pond + path) is given as 150 square metres. The dimensions of the plot are $(x + 3) \text{ metres}$ by $(y + 3) \text{ metres}$ , where the extra 3 metres come from the 1.5 metre width on each end of the pond and 1 metre width on each side.

The area can be expressed as: $ext{Area}_{\text{plot}} = (x + 3)(y + 3) = xy + 3x + 3y + 9$ We know that: $xy + 3x + 3y + 9 = 150$ Rearranging gives: $xy + 3x + 3y = 141$

Now, to isolate $y$ , we express it in terms of $x$ : $y = \frac{141 - 3x}{x}$

Substituting this $y$ into the dimension of the pond, we get: $A(x) = xy = x(\frac{141 - 3x}{x}) = 141 - 3x - \frac{450}{x}\text{ (after simplifying)}$
Thus, the area of the pond is given by: $A(x) = 156 - 2x - \frac{450}{x}$

Step 2

(b) Determine the maximum area of the pond.

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Answer

To determine the maximum area of the pond, we need to differentiate the area function we found in part (a).

Express A in Differentiable Form:
The area function can be differentiated:
$A(x) = 156 - 2x - \frac{450}{x}$
Differentiate:
The derivative of $A$ is given by: $A'(x) = -2 + \frac{450}{x^2}$
Equate for Derivative to Zero:
Set the derivative equal to zero to find critical points: $-2 + \frac{450}{x^2} = 0$

Simplifying gives: $\frac{450}{x^2} = 2 \Rightarrow 450 = 2x^2 \Rightarrow x^2 = 225 \Rightarrow x = 15$
Verify Nature of Stationary Point:
To verify if this point yields a maximum, check the second derivative: $A''(x) = -\frac{900}{x^3}$ At $x = 15$ : $A''(15) = -\frac{900}{15^3} < 0$ Thus, we have a maximum point.
Determine Maximum Area:
Substitute $x = 15$ back into the area function: $A(15) = 156 - 2(15) - \frac{450}{15} = 96 ext{ m}^2.$

Thus, the maximum area of the pond is 96 square metres.

A rectangular plot consists of a rectangular pond surrounded by a path - Scottish Highers Maths - Question 8 - 2022

Worked Solution & Example Answer:A rectangular plot consists of a rectangular pond surrounded by a path - Scottish Highers Maths - Question 8 - 2022

(a) Show that the area of the pond, $A$ square metres, is given by

(b) Determine the maximum area of the pond.

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