1) The area of the figure is the sum of the areas of the rectangle and the equilateral triangle. The area of the rectangle (with base x and length y) is given by . The area of the equilateral triangle is given by .
The sum of these areas is given by .
The perimeter of the figure is given by . Therefore, .
Now, substitute in the area formula.
Thus, the area function is a parabola whose maximum will occur at its vertex. Its its vertex can be found either graphically or algebraically.
2) f(x) is a piece-wise function, so tackle the problem piece by piece. Start with the first piece:
Find its domain and range since they correspond to the range and domain of the inverse function, respectively.
Here is the domain of f(x) (I'll let you find its range):
Then, find the inverse of by switching y and x and solving for the "new" y.
Repeat the steps for the second piece:
Here is the final answer:
3) The problem mentions the volume of a cone, so start with the formula for the volume of a cone:
You know the radius and height of the cone, so substitute these values into the volume formula.
But the volume of the cone is only half the volume if it were full, so .
The radius of the water is still 8 meters, and you just found its volume to be cubic meters. Use the volume formula to solve for the height h.