91Ó°ÊÓ

Find the volume generated by rotating about the \(x\) -axis the regions bounded by the graphs of each set of equations. $$y=x, x=0, x=2$$

Find the accumulated future value of cach continuous income stream at rate \(R(t),\) for the given time \(T\) and interest rate \(k\) compounded continuously. Round to the nearest \(\$ 10 .\) $$R(t)=550,000, \quad T=22 y r, \quad k=7 \%$$

Find the volume generated by rotating about the \(x\) -axis the regions bounded by the graphs of each set of equations. $$y=\frac{1}{\sqrt{x}}, x=1, x=4$$

Find the accumulated present value of cach continuous income stream at rate \(R(t),\) for the given time \(T\) and interest rate \(k\) connpounded continuously. $$R(t)=\$ 800,000, \quad T=20 y r, \quad k=8 \%$$

Volume of a football. A regulation football used in the National Football League is 11 in. from tip to tip and 7 in. in diameter at its thickest (the regulations allow for slight variation in these dimensions). (Source: NFL.) The shape of a football can be modeled by the function \(f(x)=-0.116 x^{2}+3.5,\) for \(-5.5 \leq x \leq 5.5\) where \(x\) is in inches. Find the volume of the football by rotating the region bounded by the graph of \(f\) about the \(x\) -axis.

Prove that the volume of a right-circular cone of height \(h\) and radius \(r\) is \(V=\frac{1}{3} \pi r^{2} h .\) (Hint: Rotate a line starting at the origin and ending at the point \((h, r)\) about the \(x\) -axis.)

Future value of an inheritance. Upon the death of his aunt, Burt receives an inheritance of \(\$ 80,000,\) which he invests for 20 yr at \(8.2 \%,\) compounded continuously. What is the future value of the inheritance?

Accumulated present value. Find the accumulated present value of an investment for which there is a perpetual continuous money flow of \(\$ 3600\) per year at an interest rate of \(7 \%,\) compounded continuously.

Trust fund. Bob and Ann MacKenzie have a new grandchild, Brenda. They want to create a trust fund for her that will yield \(\$ 250,000\) on her 24 th birthday, when she might want to start her own business. a) What lump sum would they have to deposit now at \(5.8 \%,\) compounded continuously, to achieve \(\$ 250,000 ?\) b) The amount in part (a) is more than they can afford, so they decide to invest a constant money stream of \(R(t)\) dollars per year. Find \(R(t)\) such that the accumulated future value of the continuous money stream is \(\$ 250,000,\) assuming an interest rate of \(5.8 \%\) compounded continuously.

The time to failure, \(t,\) in hours, of a machine is often exponentially distributed with a probability density function $$f(t)=k e^{-k t}, \quad 0 \leq t<\infty$$ where \(k=1 / a\) and \(a\) is the average amount of time that will pass before a failure occurs. Suppose that the average amount of time that will pass before a failure occurs is 100 hr. What is the probability that a failure will occur in 50 hr or less?