91Ó°ÊÓ

Boat Building Boat builders share an old rule of thumb for sailboats. The maximum speed \(K\) in knots is 1.35 times the square root of the length \(L\) in feet of the boat's waterline. a. A customer is planning to order a sailboat with a maximum speed of 8 knots. How long should the waterline be? b. How much longer would the waterline have to be to achieve a maximum speed of 10 knots?

Solve. Check for extraneous solutions. \((2 x+3)^{\frac{3}{4}}-3=5\)

Sales A car dealer offers a 10\(\%\) discount off the list price \(x\) for any car on the lot. At the same time, the manufacturer offers a \(\$ 2000\) rebate for each purchase of a car. a. Write a function \(f(x)\) to represent the price after the discount. b. Write a function \(g(x)\) to represent the price after the \(\$ 2000\) rebate. c. Suppose the list price of a car is \(\$ 18,000\) . Use a composite function to find the price of the car if the discount is applied before the rebate. d. Suppose the list price of a car is \(\$ 18,000\) . Use a composite function to find the price of the car if the rebate is applied before the discount.

a. The graph of \(y=\sqrt{x}\) is translated five units to the right and two units down. Write an equation of the translated function. b. The translated graph from part (a) is again translated, this time four units left and three units down. Write an equation of the translated function.

Satellites The circular velocity \(v,\) in miles per hour, of a satellite orbiting Earth is given by the formula \(v=\sqrt{\frac{1.24 \times 10^{12}}{r}, \text { where } r \text { is the distance }}\) in miles from the satellite to the center of Earth. How much greater is the velocity of a satellite orbiting at an altitude of 100 \(\mathrm{mi}\) than one orbiting at an altitude of 200 \(\mathrm{mi} ?\) (The radius of Earth is 3950 \(\mathrm{mi}\) .)

In the expression \(\sqrt[n]{x^{m}}, m\) and \(n\) are positive integers and \(x\) is a real number. The expression can be simplified. a. If \(x>0,\) what are the possible values for \(m\) and \(n\) ? b. If \(x<0,\) what are the possible values for \(m\) and \(n\) ? c. If \(x<0\) and an absolute value symbol is needed in the simplified expression, what are the possible values of \(m\) and \(n ?\)

For each function \(f,\) find \(f^{-1},\) the domain and range of \(f\) and \(f^{-1},\) and determine whether \(f^{-1}\) is a function. $$ f(x)=\frac{1}{(x+1)^{2}} $$

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=-\sqrt[3]{8 x-2}\)

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=10-\sqrt[3]{\frac{x+3}{27}}\)

Solve \(5 \sqrt{x}+7=8\)