A rectangular window is topped with a semicircle. The height of the rectangular part is 1 more than 3 times its width, w meters. Which function represents the total area, A, of the window in terms of the width?
width = w
height of rectangle = (3w+1)
area of rectangle = 3 w^2 + w
area of semicircle = (pi/2) r^2 =(pi/2)w^2/4
= pi w^2/8
total A = (3 + pi/8)w^2 + w
To find the total area of the window, we need to calculate the sum of the areas of the rectangular part and the semicircle part.
Let's start with the rectangular part. The area of a rectangle is given by multiplying its length by its width. In this case, we know that the width is 'w' meters. According to the problem, the height of the rectangular part is 1 more than 3 times its width. So, the height of the rectangular part can be expressed as (3w + 1) meters.
Therefore, the area of the rectangular part is calculated as:
Area_rectangular = length * width = (3w + 1) * w
Now, let's move on to the semicircle part. The area of a semicircle is given by half the area of a circle. The formula for the area of a circle is πr^2, where 'r' is the radius. In this case, the radius of the semicircle is equal to half the width of the rectangle (w/2).
So, the area of the semicircle is calculated as:
Area_semicircle = (1/2) * π * (w/2)^2
To find the total area A, we add the area of the rectangular part and the area of the semicircle part:
A = Area_rectangular + Area_semicircle
A = (3w + 1) * w + (1/2) * π * (w/2)^2
Simplifying this expression, we have the function representing the total area A in terms of the width w:
A = 3w^2 + w + (1/8) * πw^2
To find the total area of the window, we need to find the areas of the rectangular part and the semicircle part separately and then sum them up.
Let's first find the area of the rectangular part. The area of a rectangle is given by the formula: A_rect = length × width.
In this case, the width is given as 'w' meters. And the height of the rectangular part is 1 more than 3 times its width. So the length of the rectangle can be expressed as (3w + 1).
Therefore, the area of the rectangular part is: A_rect = w × (3w + 1).
Now let's find the area of the semicircle part. The area of a semicircle is given by the formula: A_semicircle = (π × r^2) / 2, where 'r' is the radius of the semicircle.
Since the width of the rectangle is also the diameter of the semicircle, the radius can be expressed as r = w/2.
Therefore, the area of the semicircle part is: A_semicircle = (π × (w/2)^2) / 2.
Now, to find the total area of the window, we need to add the areas of the rectangular and semicircle parts.
A_total = A_rect + A_semicircle
Substituting the expressions we derived earlier, we get:
A_total = (w × (3w + 1)) + ((π × (w/2)^2) / 2)
So, the function that represents the total area of the window in terms of the width 'w' is:
A(w) = w(3w + 1) + (π(w^2)/4)