The cost of manufacturing a semi-circular window is the cost of the glass pane together with the cost of the framing material. When ordering this window, a customer specifies the length of the base of the window, shown as l.
b. Write a function named k that relates the area of the glass pane to the length of the base of the window.
k = 1/2 π(l/2)^2 = π/8 l^2
To write a function that relates the area of the glass pane to the length of the base of the window, we need to find the formula for the area of the glass pane in terms of the length of the base.
The area of a semi-circle is given by the formula A = (π * r^2) / 2, where r is the radius of the semi-circle.
Since the length of the base of the window is given as 'l', we can use the formula for the radius of a semi-circle, r = l / 2.
Substituting the value of r in the area formula, we get:
A = (π * (l/2)^2) / 2
Simplifying the equation:
A = (π * l^2) / 8
Therefore, the function k that relates the area of the glass pane to the length of the base of the window is:
k(l) = (π * l^2) / 8