Part 1:
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.
Part 2:
Use the function you came up with in the previous part to find the combined area if the walkway is 3ft.
Area of garden itself = 20(15) ft^2 = 300 ft^2
Area of walk and garden = (20+2x)(15+2x) ft^2
area of walk alone = A(x) = (20+2x)(15+2x) - 300 ft^2
Now plug in x = 3 and evaluate
Part 1:
To find the combined area of the garden walk, we need to calculate the area of the outer rectangle and subtract the area of the inner rectangle (which represents the garden itself).
The width of the garden walk is given as 'x', and the dimensions of the garden are 20ft by 15ft.
The length of the outer rectangle will be the sum of the length of the garden and twice the width of the walk, as we have the walk along all four sides. Similarly, the width of the outer rectangle will be the sum of the width of the garden and twice the width of the walk.
So, the length of the outer rectangle becomes (20 + 2x) ft, and the width becomes (15 + 2x) ft.
The area of the outer rectangle is then given by:
Area_outer = Length_outer * Width_outer
= (20 + 2x) * (15 + 2x)
Now, we need to subtract the area of the garden (20ft by 15ft) from the area of the outer rectangle to get the combined area of the garden walk.
The area of the garden is given by:
Area_garden = Length_garden * Width_garden
= 20 * 15
Therefore, the combined area of the garden walk, A(x), can be expressed as a polynomial in standard form:
A(x) = Area_outer - Area_garden
= (20 + 2x) * (15 + 2x) - (20 * 15)
= 4x^2 + 70x
Part 2:
To find the combined area of the garden walk when the walkway width, x, is 3ft, we can substitute x = 3 into the polynomial we derived in Part 1:
A(3) = 4(3)^2 + 70(3)
= 36 + 210
= 246 ft^2
Therefore, the combined area of the garden walk when the walkway width is 3ft is 246 square feet.