1. To draw the graph of A = 24x - 2x², you can plot points by substituting different values of x into the equation and solving for A. Then, connect the points to create a smooth curve.
Here is the graph:
2. To find the largest possible area that can be fenced, we need to find the maximum point on the graph. This can be done by finding the x-coordinate of the vertex of the quadratic equation 24x - 2x².
The equation can be rewritten as A = -2x² + 24x. The x-coordinate of the vertex can be found using the formula x = -b/2a, where a = -2 and b = 24.
x = -24/(2*(-2))
x = -24/(-4)
x = 6
Plug x = 6 back into the equation A = 24x - 2x² to find the maximum area:
A = 24*6 - 2*6²
A = 144 - 2*36
A = 144 - 72
A = 72
Therefore, the largest possible area that can be fenced is 72 m².
3. The reason why x > 0 is because in the context of the problem, x represents the width of the garden. It is not possible to have a negative width. Additionally, the equation A = 24x - 2x² models the area of the garden, and an area cannot be negative. Therefore, x must be greater than 0.
4. To find the width of the garden when its area is 60 m², we can look at the graph. Find the point on the graph where A = 60, and read off the corresponding x-coordinate.
From the graph, it appears that the width of the garden when its area is 60 m² is approximately x = 3.
A villager has 24 m of fencing and builds a rectangular garden using an existing fence as one side. The area, A m², of the garden is given by A = 24x - 2x², where x is the width of the garden.
Use the information above to solve the following questions:
1. Draw the graph of A = 24x - 2x²
2. What is the largest possible area that can be fenced?
3. Why is x > 0?
4. Use the graph in question 1 to find the width of the garden when its area is 60 m.
1 answer