91Ó°ÊÓ

According to the Center of Science in the Public Interest, the maximum healthy weight for a person who is 5 feet 5 inches tall is 150 pounds, and the maximum healthy weight for someone 6 feet 3 inches tall is 200 pounds. The relationship between weight and height here is linear. (a) Find a linear equation that gives the maximum healthy weight \(y\) for a person whose height is \(x\) inches over4 feet 10 inches. (Thus \(x=0\) corresponds to 4 feet \(10 \text { inches, } x=2 \text { to } 5 \text { feet, etc. }) \quad U_{S \in}\). (b) What is the maximum healthy weight for a person whose cise height is 5 feet? 6 feet? (c) How tall is a person who is at a maximum healthy.

One diagonal of a square has endpoints (-3,1) and \((2,-4) .\) Find the endpoints of the other diagonal.

Find all points on the \(y\) -axis that are 8 units from (-2,4).

The atmospheric pressure \(a\) (in pounds per square foot) at height \(h\) thousand feet above sea level is approximately $$ a=8315 h^{2}-73.93 h+2116.1 $$ (a) Find the atmospheric pressure at sea level and at the top of Mount Everest, the tallest mountain in the world \(\left(29,035 \text { feet }^{*}\right) .\) [Remember that \(h\) is measured in thousands.] (b) The atmospheric pressure at the top of Mount Rainier is 1223.43 pounds per square foot. How high is Mount Rainier?

Show that the diagonals of a rectangle have the same length. [Hint: Place the rectangle in the first quadrant of the plane and label its vertices appropriately, as in Exercises \(89-90 .\) ]

If the diagonals of a parallelogram have the same length, show that the parallelogram is actually a rectangle. [Hint: See Exercise 90.1

Galileo discovered that the period of a pendulum depends only on the length of the pendulum and the acceleration of gravity. The period \(T\) of a pendulum (in seconds) is $$T=2 \pi \sqrt{\frac{l}{g}}$$ where \(l\) is the length of the pendulum in feet and \(g \approx\) 32.2 \(\mathrm{ft} / \mathrm{sec}^{2}\) is the acceleration due to gravity. Find the period of a pendulum whose length is 4 feet.

Show that the diagonals of a square are perpendicular. I Hint: Place the square in the first quadrant of the plane, with one vertex at the origin and sides on the positive axes. Label the coordinates of the vertices appropriately.]

Proof of the Midpoint Formula Let \(P\) and \(Q\) be the points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right),\) respectively, and let \(M\) be the point with coordinates $$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$$ Use the distance formula to compute the following: (a) The distance \(d\) from \(P\) to \(Q\) (b) The distance \(d_{1}\) from \(M\) to \(P\) (c) The distance \(d_{2}\) from \(M\) to \(Q\) (d) Verify that \(d_{1}=d_{2}\) (e) Show that \(d_{1}+d_{2}=d .\) [Hint: Verify that \(d_{1}=\frac{1}{2} d\) and \(\left.d_{2}=\frac{1}{2} d .\right]\) (f) Explain why parts (d) and (e) show that \(M\) is the midpoint of \(P Q\).

Find a number \(k\) such that the given equation has exactly one real solution. $$k x^{2}+8 x+1=0$$