Final answer:
To create a fence that encloses the maximum area of the garden, we need to find the values of x and y. Using the given information and equations, we can solve for x and y. The values that maximize the area are x = 1.8 feet and y = 4.4 feet.
Step-by-step explanation:
To find the values of x and y that will create a fence that encloses the maximum area of the garden, we need to use the given information. We have 36 feet of wire to enclose 3 sides of the garden, and each vertical side requires 5 strands of wire while the horizontal side requires 3 strands of wire. Let's assume the length of each vertical side is x and the length of the horizontal side is y.
From the information given, we can form the equation: 5x + 5x + 3y = 36. Simplifying this equation, we get 10x + 3y = 36.
Since we want to maximize the area of the garden, we can use the formula for the area of a rectangle: A = xy. We need to express y in terms of x.
Let's solve the equation 10x + 3y = 36 for y: 3y = 36 - 10x. Dividing both sides by 3, we get y = (36 - 10x)/3.
Substituting this expression for y into the area formula, we have A = x((36 - 10x)/3). To find the values of x and y that maximize the area, we can graph the equation and find the maximum point, or we can take the derivative with respect to x and set it equal to 0.
Taking the derivative of A with respect to x, we get dA/dx = (36 - 20x)/3.
Setting this equal to 0, we have (36 - 20x)/3 = 0.
Solving for x, we find x = 36/20
= 1.8.
Plugging this value back into the equation for y, we get y = (36 - 10(1.8))/3
= 4.4.
Therefore, the values of x and y that will create a fence that encloses the maximum area of the garden are x = 1.8 feet and y = 4.4 feet.