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. 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.
To solve these questions, we will go step by step.
1. To draw the graph of A = 24x - 2x², we can plot values of A for different values of x. Let's choose some values to start with:
When x = 0, A = 24(0) - 2(0)² = 0
When x = 4, A = 24(4) - 2(4)² = 16
When x = 6, A = 24(6) - 2(6)² = 12
Plotting these points on a graph will give you a downward-opening parabola. Make sure to label the axes and units of measurement (A in m² and x in m).
2. To find the largest possible area that can be fenced, we need to find the maximum value of A in the equation A = 24x - 2x². This can be done by finding the vertex of the parabola.
The vertex of a downward-opening parabola given by the equation y = ax² + bx + c is given by the coordinates (h, k), where h = -b/(2a) and k = f(h).
In this case, a = -2, b = 24, and c = 0 (because there is no constant term). Therefore, h = -24/(2(-2)) = 6, and k = f(6) = 12.
So, the maximum area that can be fenced is 12 m².
3. The inequality x > 0 is necessary because in this context, x represents the width of the garden. It does not make sense to have a negative width. The garden needs to have a positive width in order to exist.
4. To find the width of the garden when its area is 60 m², we can refer back to the graph.
Locate the point on the graph where A = 60. From the graph, you can determine the corresponding value of x. It appears to be between 2 and 4.
To find the exact value, you can use algebra. Set A = 60 in the equation A = 24x - 2x² and solve for x. You will obtain a quadratic equation: 2x² - 24x + 60 = 0. Solve this equation for x using factoring, completing the square, or the quadratic formula.
The two solutions to the quadratic equation will give you two possible widths for the garden. Choose the one that makes sense in the context of the problem.
1. To draw the graph of A = 24x - 2x², we can plot points on a coordinate grid to represent different values of x and calculate the corresponding values of A.
Here are a few values for x that we can use:
- x = 0: A = 0
- x = 4: A = 48
- x = 6: A = 84
- x = 10: A = 140
Plotting these points on the graph and connecting them, we get a downward parabolic curve.
2. To find the largest possible area that can be fenced, we need to find the maximum value of A. Since A is a quadratic equation in terms of x, the maximum value occurs at the vertex of the parabola.
The x-coordinate of the vertex can be found using the formula x = -b/2a, where a = -2 and b = 24 in this case.
x = -24 / (2 * (-2)) = 6
Substituting this value of x back into the equation, we can find the maximum area:
A = 24(6) - 2(6)² = 144 - 72 = 72 m²
Therefore, the largest possible area that can be fenced is 72 square meters.
3. In this context, x represents the width of the garden. To make physical sense, the width cannot be negative. Since x represents a length, it must be greater than zero.
4. Using the graph from question 1, we can find the width of the garden when its area is 60 m². We look for the point on the graph where A = 60.
From the graph, we can estimate that the width of the garden is approximately x = 7.5 when the area is 60 m².