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 | 4.5 | 3.5 | 2.5 | 1.5 | 0.5 | - | - | - | - | | A (m²) | 26 | - | - | - | - | - | - | - | - | - | - | (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 Calculus - Differentiation: 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

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 can set up the equation:

$2x + 2y = 21$

Dividing through by 2 gives:

$x + y = 10.5$

Rearranging this gives:

$y = 10.5 - x$

Thus, we have shown that ( y = 10.5 - x ).

Step 2

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 find the area A of the rectangle, we use the formula:

$A = x imes y$

Substituting for y gives:

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

Now, substitute the values of x from 0 to 10 into this formula to complete the table:

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 )

This can be filled into the provided table.

Step 3

Use the graph to estimate the maximum value of A and write the corresponding length and width.

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Answer

From the graph (which should be plotted based on the values from the previous step), observe that the maximum area occurs at approximately x = 5.25.

At this value of x:

The corresponding value of y can be estimated as ( y = 10.5 - 5.25 = 5.25 ).
Hence, the maximum area is approximately ( A = 27.56 m^2 ).

Step 4

Show that the area of the rectangle can be written as A = 10.5x - x².

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Answer

As previously derived, we can express the area A of the rectangle as:

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

This confirms the area formula.

Step 5

Find $ \frac{dA}{dx} $.

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Answer

Differentiating the area function with respect to x gives:

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

Step 6

Hence, find the value of x which will give the maximum area.

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Answer

To find the critical points where possible maximum areas occur, set the derivative equal to zero:

$10.5 - 2x = 0$

Solving for x yields:

$2x = 10.5 \Rightarrow x = 5.25$

Step 7

Find this maximum area.

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Answer

Substituting x = 5.25 back into the area formula:

$A = 10.5(5.25) - (5.25)^2 = 27.56 m^2$

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

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.

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

Use the graph to estimate the maximum value of A and write the corresponding length and width.

Show that the area of the rectangle can be written as A = 10.5x - x².

Find \( \frac{dA}{dx} \).

Hence, find the value of x which will give the maximum area.

Find this maximum area.

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