Mr. Bigelow's garden is a rectangle with dimensions x feet long and y feet wide. Its perimeter is 70 feet.
a. Write a function to represent the perimeter of his garden.
b. What are the x- and y-intercepts of the function?
c. Does either intercept make sense as a solution for this situation?
Explain.
Can someone teach me how to do this? I don't understand.
Sure! Let's break down the problem step-by-step.
a. Write a function to represent the perimeter of Mr. Bigelow's garden.
The perimeter of a rectangle is calculated by adding up the lengths of all four sides. In this case, the perimeter is given as 70 feet. The dimensions of the garden are given as x feet long and y feet wide.
To calculate the perimeter, we can use the formula: P = 2x + 2y.
b. What are the x- and y-intercepts of the function?
The x-intercept of a function is the point where the graph intersects the x-axis (y = 0), and the y-intercept is the point where the graph intersects the y-axis (x = 0).
To find the x-intercept, we set y = 0 and solve for x:
P = 2x + 2(0)
70 = 2x
x = 35
So the x-intercept is (35, 0).
To find the y-intercept, we set x = 0 and solve for y:
P = 2(0) + 2y
70 = 2y
y = 35
So the y-intercept is (0, 35).
c. Does either intercept make sense as a solution for this situation? Explain.
In this situation, the intercepts represent points where one side of the garden is zero. However, in a real-world scenario, it wouldn't make sense for a side of the garden to have a length of zero. Therefore, neither intercept (0, 35) nor (35, 0) makes sense as a solution to this particular situation.
Sure! Let's break down each part of the question and work through it step by step.
a. To represent the perimeter of Mr. Bigelow's garden, we can use the formula for the perimeter of a rectangle, which is given by: P = 2*(length + width). In this case, the length is x feet and the width is y feet, so the function to represent the perimeter would be: P(x, y) = 2*(x + y).
b. To find the x-intercept of the function, we need to set y = 0 and solve for x. Let's plug in y = 0 into the perimeter function: P(x, 0) = 2*(x + 0) = 2x. Since the perimeter can't be negative, the x-intercept is (0, 0), which means that if the width is 0, the garden has no perimeter.
To find the y-intercept, we set x = 0 and solve for y. Plugging x = 0 into the perimeter function, we get: P(0, y) = 2*(0 + y) = 2y. Again, since the perimeter can't be negative, the y-intercept is (0, 0), which means that if the length is 0, the garden has no perimeter.
c. Now, let's consider if either intercept makes sense as a solution in this situation. The x-intercept (0, 0) represents a garden with no width, which means it would be a line of zero width, rather than a rectangular garden. This does not make sense in the context of a garden.
Similarly, the y-intercept (0, 0) represents a garden with no length, which would also be a line of zero length and not a proper rectangular garden. Therefore, neither intercept makes sense as a solution for this situation.
In conclusion, the function P(x, y) = 2*(x + y) represents the perimeter of Mr. Bigelow's garden, but the x- and y-intercepts (0, 0) do not make sense in this context.
2x+2y = 70
an intercept is when the other variable is zero. Having length or width = 0 isn't too useful.