Chapter 9

Use Cramer's rule to solve the system for \(x\). $$ \left\\{\begin{array}{l} a x+b y+c z=d \\ e x+f z=g \\ h x+i y=j \end{array}\right. $$

Forest growth Temperature and precipitation have a significant effect on plant life. If either the average annual temperature or the amount of precipitation is too low, trees and forests cannot grow. Instead, only grasslands and deserts will exist. The relationship between average annual temperature \(T\) (in \({ }^{\circ} \mathrm{F}\) ) and average annual precipitation \(P\) (in inches) is a linear inequality. In order for forests to grow in a region, \(T\) and \(P\) must satisfy the inequality \(29 T-39 P<450\), where \(33 \leq T \leq 80\) and \(13 \leq P \leq 45\). (a) Determine whether forests can grow in Winnipeg, where \(T=37^{\circ} \mathrm{F}\) and \(P=21.2 \mathrm{in} .\) (b) Graph the inequality, with \(T\) on the horizontal axis and \(P\) on the vertical axis. (c) Identify the region on the graph that represents where forests can grow.

An avid tennis watcher wants to record 6 hours of a major tournament on a single tape. Her tape can hold 5 hours and 20 minutes at the LP speed and 8 hours at the slower SLP speed. The LP speed produces a better quality picture, so she wishes to maximize the time recorded at the LP speed. Find the amount of time to be recorded at each speed.

Suppose consumers will buy \(1,000,000\) T-shirts if the selling price is $$\$ 15$$, but for each $$\$ 1$$ increase in price, they will buy 100,000 fewer T-shirts. Moreover, suppose vendors will order \(2,000,000\) T-shirts if the selling price is $$\$ 20$$, and for every $$\$ 1$$ increase in price, they will order an additional 150,000 . (a) Express the number \(Q\) of T-shirts consumers will buy if the selling price is \(p\) dollars. (b) Express the number \(K\) of T-shirts vendors will order if the selling price is \(p\) dollars. (c) Determine the market price-that is, the price when \(Q=K\).

A competition model is a collection of equations that specifies how two or more species interact in competition for the food resources of an ecosystem. Let \(x\) and \(y\) denote the numbers (in hundreds) of two competing species, and suppose that the respective rates of growth \(R_{1}\) and \(R_{2}\) are given by $$ \begin{aligned} &R_{1}=0.01 x(50-x-y) \\ &R_{2}=0.02 y(100-y-0.5 x) \end{aligned} $$ Determine the population levels \((x, y)\) at which both rates of growth are zero. (Such population levels are called stationary points.)

Exer. 45-48: Solve the system for \(a\) and \(b\). (Hint: Treat terms such as \(e^{3 x}, \cos x\), and \(\sin x\) as "constant coefficients.") $$ \left\\{\begin{array}{c} a e^{3 x}+b e^{-3 x}=0 \\ a\left(3 e^{3 x}\right)+b\left(-3 e^{-3 x}\right)=e^{3 x} \end{array}\right. $$

A rancher has 2420 feet of fence to enclose a rectangular region that lies along a straight river. If no fence is used along the river (see the figure), is it possible to enclose 10 acres of land? Recall that 1 acre \(=43,560 \mathrm{ft}^{2}\). Exercise 46

Exer. 45-48: Solve the system for \(a\) and \(b\). (Hint: Treat terms such as \(e^{3 x}, \cos x\), and \(\sin x\) as "constant coefficients.") $$ \left\\{\begin{array}{r} a e^{-x}+b e^{4 x}=0 \\ -a e^{-x}+b\left(4 e^{4 x}\right)=2 \end{array}\right. $$

Exer. 45-48: Solve the system for \(a\) and \(b\). (Hint: Treat terms such as \(e^{3 x}, \cos x\), and \(\sin x\) as "constant coefficients.") $$ \left\\{\begin{aligned} a \cos x+b \sin x &=0 \\ -a \sin x+b \cos x &=\tan x \end{aligned}\right. $$

The isoperimetric problem is to prove that of all plane geometric figures with the same perimeter (isoperimetric figures), the circle has the greatest area. Show that no rectangle has both the same area and the same perimeter as any circle.

Exer. 45-48: Solve the system for \(a\) and \(b\). (Hint: Treat terms such as \(e^{3 x}, \cos x\), and \(\sin x\) as "constant coefficients.") $$ \left\\{\begin{array}{c} a \cos x+b \sin x=0 \\ -a \sin x+b \cos x=\sin x \end{array}\right. $$

A moiré pattern is formed when two geometrically regular patterns are superimposed. Shown in the figure is a pattern obtained from the family of circles \(x^{2}+y^{2}=n^{2}\) and the family of horizontal lines \(y=m\) for integers \(m\) and \(n\). (a) Show that the points of intersection of the circle \(x^{2}+y^{2}=n^{2}\) and the line \(y=n-1\) lie on a parabola. (b) Work part (a) using the line \(y=n-2\). Exercise 49

A spherical pill has diameter 1 centimeter. A second pill in the shape of a right circular cylinder is to be manufactured with the same volume and twice the surface area of the spherical pill. (a) If \(r\) is the radius and \(h\) is the height of the cylindrical pill, show that \(6 r^{2} h=1\) and \(r^{2}+r h=1\). Conclude that \(6 r^{3}-6 r+1=0\). (b) The positive solutions of \(6 r^{3}-6 r+1=0\) are approximately \(0.172\) and \(0.903\). Find the corresponding heights, and interpret these results.

Hammer throw A hammer thrower is working on his form in a small practice area. The hammer spins, generating a circle with a radius of 5 feet, and when released, it hits a tall screen that is 50 feet from the center of the throwing area. Let coordinate axes be introduced as shown in the figure (not to scale). (a) If the hammer is released at \((-4,-3)\) and travels in the tangent direction, where will it hit the screen? (b) If the hammer is to hit at \((0,-50)\), where on the circle should it be released? Exercise 51