A rectangular garden has a walk around it of a width x. The garden is 20 ft by 15 ft. Write a function representing the combined area, A(x), of the garden walk. Write as a polynomial in standard form.
Use the function you came up with in the previous question to find the combined area if the walkway is 3ft. Write a sentence that gives your labeled answer.
since there is a walkway on both sides, each dimension is increased by 2x.
A(x) = (20+2x)(15+2x)
To find the combined area of the garden walk, we need to calculate the area of the rectangular garden and subtract the area of the inner rectangle. The outer rectangle is formed by increasing the length and width of the garden by twice the width of the walkway.
Let's denote the width by x. The length and width of the outer rectangle are (20 + 2x) and (15 + 2x) respectively. The area of the outer rectangle is given by:
A_outer = (20 + 2x) * (15 + 2x)
Next, we need to calculate the area of the inner rectangle. The length and width of the inner rectangle are the original length and width of the garden: 20 ft and 15 ft respectively. Therefore, the area of the inner rectangle is:
A_inner = 20 * 15
Finally, we can find the combined area by subtracting the area of the inner rectangle from the area of the outer rectangle:
A(x) = A_outer - A_inner
= (20 + 2x) * (15 + 2x) - (20 * 15)
Now, let's substitute x = 3 to find the combined area when the walkway width is 3 ft:
A(3) = (20 + 2*3) * (15 + 2*3) - (20 * 15)
= 26 * 21 - 300
= 546 - 300
= 246
Therefore, when the walkway width is 3 ft, the combined area of the garden walk is 246 square feet.