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) | 10.5 | 9.5 | 8.5 | 7.5 | 6.5 | 5.5 | 4.5 | 3.5 | 2.5 | 1.5 | 0.5 | | A (m²) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | (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 (m²) Length (m) Width (m) (d) (i) Show that the area of the rectangle can be written as $A = 10.5x - x²$. (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 Area & Volume: 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 have:

$2x + 2y = 21$

Dividing the entire equation by 2 gives:

$x + y = 10.5$

Rearranging the equation for y gives:

$y = 10.5 - x$

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 compute the area A of the rectangle, we can use the formula:

$A = xy$

Substituting the expression for y into this, we have:

$A = x(10.5 - x)$

Now we can calculate y and A for the values of x from 0 to 10:

For x = 0: y = 10.5, A = 0
For x = 1: y = 9.5, A = 9.5
For x = 2: y = 8.5, A = 17
For x = 3: y = 7.5, A = 22.5
For x = 4: y = 6.5, A = 26
For x = 5: y = 5.5, A = 27.5
For x = 6: y = 4.5, A = 27
For x = 7: y = 3.5, A = 24.5
For x = 8: y = 2.5, A = 20
For x = 9: y = 1.5, A = 13.5
For x = 10: y = 0.5, A = 5

Thus, we get the complete table.

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 of A against x, plot the points from the table:

(0, 0)
(1, 9.5)
(2, 17)
(3, 22.5)
(4, 26)
(5, 27.5)
(6, 27)
(7, 24.5)
(8, 20)
(9, 13.5)
(10, 5)

Join the points smoothly to form a parabolic curve.

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

From the graph, the maximum value of A appears to be approximately 27.5 m² at x = 5 m.

Thus, the corresponding values are:

Maximum area: 27.5 m²
Length: 5 m
Width: 5 m

Step 5

(d) (i) Show that the area of the rectangle can be written as $A = 10.5x - x²$.

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Answer

Using the previous equation for A:

$A = x(10.5 - x)$

Distributing x yields:

$A = 10.5x - x^2$

Step 6

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

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Answer

To find the derivative of A, we differentiate:

$\frac{dA}{dx} = 10.5 - 2x$

Step 7

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

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Answer

To find the critical points, set the derivative equal to zero:

$10.5 - 2x = 0$

Solving for x gives:

$x = 5.25$

Step 8

(d) (iv) Find this maximum area.

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Answer

Substituting $x = 5.25$ back into the equation for A:

$A = 10.5(5.25) - (5.25)^2$

Calculating this yields:

$A = 27.5625 ext{ m²}$

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 $A = 10.5x - x²$.

(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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