A rectangular garden has a walk around it of width x. The garden is 20 ft by 15 ft. Write a function representing the combined area, A(x), of the garden and walk. Write as a polynomial is standard form.
Draw a diagram. That simple step will make solution a lot easier.
A(x) = (20+2x)(15+2x)
Now you can expand that to standard form.
To find the combined area of the garden and walk, we need to add the area of the garden to the area of the walkway.
The width of the walkway is given as x. This means that the length of the garden including the walkway on both sides would be 20 + 2x, and the width including the walkway on both ends would be 15 + 2x.
The area of the garden is determined by multiplying the length and width:
Area of the garden = (20 + 2x) * (15 + 2x)
So the area of the walkway alone would be:
Area of the walkway = (20 + 2x) * (15 + 2x) - 20 * 15
To put it in standard form, we expand and simplify the equation:
Area of the walkway = 20 * 15 + 2x * 20 + 2x * 15 + 2x * 2x + 2x * 2x - 20 * 15
Simplifying further:
Area of the walkway = 300 + 40x + 30x + 4x^2 + 4x^2 - 300
Combining like terms and removing the unnecessary terms:
Area of the walkway = 80x + 8x^2
Therefore, the function representing the combined area, A(x), of the garden and walkway in standard form is:
A(x) = 80x + 8x^2