Kieran has 21 metres of fencing. He wants to enclose a vegetable garden in a rectangular shape as shown. (a) By writing an expression for the perimeter of the vegetable garden in terms of $x$ (length in metres) and $y$ (width in metres), show that $y = 10 - 5 - x$. (b) (i) Complete the table below to show the values of $y$ and $A$ (the area of the garden) for each given value of $x$. | $x$ (m) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |----------|---|---|---|---|---|---|---|---|---|---|----| | $y$ (m) | 6.5 | 5.5 | 5 | 4.5 | 4 | 3.5 | 3 | 2.5 | 2 | 1.5 | 0.5 | | $A$ (m$^2$) | 26 | 9.5 | 10 | 13.5 | 16 | 17.5 | 18 | 17.5 | 16 | 13.5 | 5 | (b) (ii) Use the values of $x$ and $A$ from the table to plot the graph of $A$ on the grid below. (c) Use your graph to estimate the maximum value of $A$ and write the corresponding length and width. | $A$: Maximum area (m$^2$) | Length (m) | Width (m) | |------------------------------|-------------|------------| (d) (i) Show that the area of the rectangle can be written as $A = 10.5 - x^2$. (ii) Find $\frac{dA}{dx}$. (iii) Hence, find the value of $x$ which will give the maximum area. (iv) Find this maximum area.

Leaving Cert Mathematics Graphs of Functions: Kieran has 21 metres of fencing.

Kieran has 21 metres of fencing - Leaving Cert Mathematics - Question 8 - 2016

Question 8

Kieran has 21 metres of fencing. He wants to enclose a vegetable garden in a rectangular shape as shown. (a) By writing an expression for the perimeter of the veget... show full transcript

Worked Solution & Example Answer:Kieran has 21 metres of fencing - Leaving Cert Mathematics - Question 8 - 2016

Step 1

a) By writing an expression for the perimeter of the vegetable garden in terms of $x$ (length in metres) and $y$ (width in metres), show that $y = 10 - 5 - x$.

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Answer

The perimeter (P) of a rectangle is given by the formula:

$P = 2x + 2y$

Given that Kieran has 21 metres of fencing, we set the perimeter equation to 21:

$2x + 2y = 21$

Dividing the entire equation by 2 yields:

$x + y = 10.5$

Now, we need to express $y$ in terms of $x$ :

$y = 10.5 - x$

This leads to your question statement. Since the problem states $y = 10 - 5 - x$ , it appears to be a reduction from the previously derived equation.

Step 2

b (i) Complete the table below to show the values of $y$ and $A$ (the area of the garden) for each given value of $x$.

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Answer

To complete the table, we calculate each corresponding value of $y$ using the equation $y = 10.5 - x$ and then calculate the area $A$ :

For $x = 0$ : $y = 10.5 - 0 = 10.5$ , hence $A = x imes y = 0 imes 10.5 = 0$ .
For $x = 1$ : $y = 10.5 - 1 = 9.5$ , hence $A = 1 imes 9.5 = 9.5$ .
Continue for each value of $x$ ...

The completed table will show:

$x$ (m)	0	1	2	3	4	5	6	7	8	9	10
$y$ (m)	10.5	9.5	8.5	7.5	6.5	5.5	4.5	3.5	2.5	1.5	0.5
$A$ (m $^2$ )	0	9.5	17	22.5	27	30	31.5	30	27	22.5	0

Step 3

b (ii) Use the values of $x$ and $A$ from the table to plot the graph of $A$ on the grid below.

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Answer

To plot the graph, use the values of $x$ from the completed table on the horizontal axis and the corresponding $A$ values on the vertical axis. Mark the points and attempt to connect them smoothly to show the area as a quadratic function. Ensure the graph illustrates the parabolic nature typical of quadratic area functions.

Step 4

c) Use your graph to estimate the maximum value of $A$ and write the corresponding length and width.

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Answer

Upon examining the graph, the maximum area occurs at the vertex of the parabola. Let's estimate the maximum area, say around $A = 27.5 ext{ m}^2$ , which corresponds to $x ext{ (length) } = 5 ext{ m }$ and $y ext{ (width) } = 5 ext{ m }$ as obtained from the equation $y = 10.5 - x$ .

Step 5

d (i) Show that the area of the rectangle can be written as $A = 10.5 - x^2$.

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Answer

The area $A$ of a rectangle is given by the formula:

$A = xy$

Substituting $y = 10.5 - x$ into the equation gives:

$A = x(10.5 - x) = 10.5x - x^2$

Thus, we rearrange this to framing it consistent with the form $A = -x^2 + 10.5x$ .

Step 6

d (ii) Find $\frac{dA}{dx}$.

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Answer

To find the derivative of the area function, we differentiate:

$\frac{dA}{dx} = \frac{d}{dx}(-x^2 + 10.5x) = -2x + 10.5$ .

Step 7

d (iii) Hence, find the value of $x$ which will give the maximum area.

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Answer

For maximizing the area, set the derivative to zero:

$-2x + 10.5 = 0$

Solving for $x$ , we have:

x = 5.25.$$

Step 8

d (iv) Find this maximum area.

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Answer

To find the maximum area, substitute $x = 5.25$ back into the area formula:

= 55.125 - 27.5625 \ = 27.5625 ext{ m}^2.$$ Thus, the maximum area is approximately 27.56 m².

Kieran has 21 metres of fencing - Leaving Cert Mathematics - Question 8 - 2016

Worked Solution & Example Answer:Kieran has 21 metres of fencing - Leaving Cert Mathematics - Question 8 - 2016

a) By writing an expression for the perimeter of the vegetable garden in terms of $x$ (length in metres) and $y$ (width in metres), show that $y = 10 - 5 - x$.

b (i) Complete the table below to show the values of $y$ and $A$ (the area of the garden) for each given value of $x$.

b (ii) Use the values of $x$ and $A$ from the table to plot the graph of $A$ on the grid below.

c) Use your graph to estimate the maximum value of $A$ and write the corresponding length and width.

d (i) Show that the area of the rectangle can be written as $A = 10.5 - x^2$.

d (ii) Find $\frac{dA}{dx}$.

d (iii) Hence, find the value of $x$ which will give the maximum area.

d (iv) Find this maximum area.

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