Chapter 9

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{l} y=\frac{4}{x+2} \\ y=x+5 \end{array}\right. $$

\(\left\\{\begin{array}{l}|x| \geq 4 \\ |y| \geq 3\end{array}\right.\)

Solve the system. $$ \left\\{\begin{array}{l} \frac{2}{x}+\frac{3}{y}=-2 \\ \frac{4}{x}-\frac{5}{y}=1 \end{array} \quad\left(\text { Hint: Let } u=\frac{1}{x} \text { and } v=\frac{1}{y} .\right)\right. $$

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{3 x^{3}+13 x-1}{\left(x^{2}+4\right)^{2}} $$

\(\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=-\left|\begin{array}{ll}b & a \\ d & c\end{array}\right|\)

A hospital dietician wishes to prepare a corn-squash vegetable dish that will provide at least 3 grams of protein and cost no more than \(36 \not\) per serving. An ounce of creamed corn provides \(\frac{1}{2}\) gram of protein and costs \(4 \phi\). An ounce of squash supplies \(\frac{1}{4}\) gram of protein and costs \(3 \phi\). For taste, there must be at least 2 ounces of corn and at least as much squash as corn. It is important to keep the total number of ounces in a serving as small as possible. Find the combination of corn and squash that will minimize the amount of ingredients used per serving.

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{l} y=\frac{10}{x+3} \\ y=-x+8 \end{array}\right. $$

\(\left\\{\begin{array}{l}|x+2| \leq 1 \\ |y-3|<5\end{array}\right.\)

Solve the system. $$ \left\\{\begin{array}{l} \frac{3}{x-1}+\frac{4}{y+2}=2 \\ \frac{6}{x-1}-\frac{7}{y+2}=-3 \end{array}\right. $$

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{2 x^{4}-2 x^{3}+6 x^{2}-5 x+1}{x^{3}-x^{2}+x-1} $$

A contractor has a large building that she wishes to convert into a series of rental storage spaces. She will construct basic \(8 \mathrm{ft} \times 10 \mathrm{ft}\) units and deluxe \(12 \mathrm{ft} \times 10 \mathrm{ft}\) units that contain extra shelves and a clothes closet. Market considerations dictate that there be at least twice as many basic units as deluxe units and that the basic units rent for $$\$ 75$$ per month and the deluxe units for $$\$ 120$$ per month. At most \(7200 \mathrm{ft}^{2}\) is available for the storage spaces, and no more than $$\$ 80,000$$ can be spent on construction. If each basic unit will cost $$\$ 800$$ to make and each deluxe unit will cost $$\$ 1600$$, how many units of each type should be constructed to maximize monthly revenue?

Find \(A B\) $$ A=\left[\begin{array}{rr} 4 & -2 \\ 0 & 3 \\ -7 & 5 \end{array}\right], \quad B=\left[\begin{array}{l} 3 \\ 4 \end{array}\right] $$

Mixing acid solutions Three solutions contain a certain acid. The first contains \(10 \%\) acid, the second \(30 \%\), and the third \(50 \%\). A chemist wishes to use all three solutions to obtain a 50 -liter mixture containing \(32 \%\) acid. If the chemist wants to use twice as much of the \(50 \%\) solution as of the \(30 \%\) solution, how many liters of each solution should be used?

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{l} y=20 / x^{2} \\ y=9-x^{2} \end{array}\right. $$

\(\left\\{\begin{aligned} x^{2}+y^{2} & \leq 4 \\ x+y & \geq 1 \end{aligned}\right.\)

The price of admission to a high school play was $$\$ 3.00$$ for students and $$\$ 4.50$$ for nonstudents. If 450 tickets were sold for a total of $$\$ 1555.50$$, how many of each kind were purchased?

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{x^{3}}{x^{3}-3 x^{2}+9 x-27} $$

\(\left|\begin{array}{cc}a & b \\ k c & k d\end{array}\right|=k\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|\)

A moose feeding primarily on tree leaves and aquatic plants is capable of digesting no more than 33 kilograms of these foods daily. Although the aquatic plants are lower in energy content, the animal must eat at least 17 kilograms to satisfy its sodium requirement. A kilogram of leaves provides four times as much energy as a kilogram of aquatic plants. Find the combination of foods that maximizes the daily energy intake.

Find \(A B\) $$ A=\left[\begin{array}{r} 4 \\ -3 \\ 2 \end{array}\right], \quad B=\left[\begin{array}{ll} 5 & 1 \end{array}\right] $$

Filling a pool A swimming pool can be filled by three pipes, A, B, and C. Pipe A alone can fill the pool in 8 hours. If pipes \(\mathrm{A}\) and \(\mathrm{C}\) are used together, the pool can be filled in 6 hours; if \(\mathrm{B}\) and \(\mathrm{C}\) are used together, it takes 10 hours. How long does it take to fill the pool if all three pipes are used?

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{aligned} x &=y^{2}-4 y+5 \\ x-y &=1 \end{aligned}\right. $$

\(\left\\{\begin{aligned} x^{2}+y^{2} & \leq 4 \\ x+y & \geq 1 \end{aligned}\right.\)

23 Ticket sales The price of admission to a high school play was $$\$ 3.00$$ for students and $$\$ 4.50$$ for nonstudents. If 450 tickets were sold for a total of $$\$ 1555.50$$, how many of each kind were purchased? 24 Air travel An airline that flies from Los Angeles to Albuquerque with a stopover in Phoenix charges a fare of $$\$ 90$$ to Phoenix and a fare of $$\$ 120$$ from Los Angeles to Albuquerque. A total of 185 passengers boarded the plane in Los Angeles, and fares totaled $$\$ 21,000$$. How many passengers got off the plane in Phoenix?

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{3 x^{2}-16}{x^{2}-4 x} $$

Find \(A B\) $$ A=\left[\begin{array}{rrrr} 2 & 1 & 0 & -3 \\ -7 & 0 & -2 & 4 \end{array}\right], \quad B=\left[\begin{array}{rrr} 4 & -2 & 0 \\ 1 & 1 & -2 \\ 0 & 0 & 5 \\ -3 & -1 & 0 \end{array}\right] $$

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{aligned} y^{2}-4 x^{2} &=4 \\ 9 y^{2}+16 x^{2} &=140 \end{aligned}\right. $$

\(\left\\{\begin{array}{l}x^{2}+y^{2}>1 \\ x^{2}+y^{2}<4\end{array}\right.\)

A crayon 8 centimeters in length and 1 centimeter in diameter will be made from \(5 \mathrm{~cm}^{3}\) of colored wax. The crayon is to have the shape of a cylinder surmounted by a small conical tip (see the figure). Find the length \(x\) of the cylinder and the height \(y\) of the cone.

Show that $$ \left|\begin{array}{lll} 1 & 1 & 1 \\ a & b & c \\ a^{3} & b^{3} & c^{3} \end{array}\right|=(a-b)(b-c)(c-a)(a+b+c) . $$

Find \(A B\) $$ A=\left[\begin{array}{rrr} 1 & 2 & -3 \\ 4 & -5 & 6 \end{array}\right], \quad B=\left[\begin{array}{rrrr} 1 & -1 & 0 & 2 \\ -2 & 3 & 1 & 0 \\ 0 & 4 & 0 & -3 \end{array}\right] $$

Electrical resistance In electrical circuits, the formula \(1 / R=\left(1 / R_{1}\right)+\left(1 / R_{2}\right)\) is used to find the total resistance \(R\) if two resistors \(R_{1}\) and \(R_{2}\) are connected in parallel. Given three resistors, A, B, and \(\mathrm{C}\), suppose that the total resistance is 48 ohms if A and B are connected in parallel, 80 ohms if \(\mathrm{B}\) and \(\mathrm{C}\) are connected in parallel, and \(60 \mathrm{ohms}\) if \(\mathrm{A}\) and \(\mathrm{C}\) are connected in parallel. Find the resistances of A, B, and C.

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{aligned} 25 y^{2}-16 x^{2} &=400 \\ 9 y^{2}-4 x^{2} &=36 \end{aligned}\right. $$

\(\left\\{\begin{array}{l}x-y^{2}<0 \\ x+y^{2}>0\end{array}\right.\)

A man rows a boat 500 feet upstream against a constant current in 10 minutes. He then rows 300 feet downstream (with the same current) in 5 minutes. Find the speed of the current and the equivalent rate at which he can row in still water.

If $$ A=\left[\begin{array}{rrrr} a_{11} & a_{12} & a_{13} & a_{14} \\ 0 & a_{22} & a_{23} & a_{24} \\ 0 & 0 & a_{33} & a_{34} \\ 0 & 0 & 0 & a_{44} \end{array}\right] $$ show that \(|A|=a_{11} a_{22} a_{33} a_{44}\)

\(\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|+\left|\begin{array}{ll}a & e \\ c & f\end{array}\right|=\left|\begin{array}{ll}a & b+e \\ c & d+f\end{array}\right|\)

Let \(A=\left[\begin{array}{rr}1 & 2 \\ 0 & -3\end{array}\right], \quad B=\left[\begin{array}{rr}2 & -1 \\ 3 & 1\end{array}\right], \quad C=\left[\begin{array}{rr}3 & 1 \\ -2 & 0\end{array}\right]\) $$ (A+B)(A-B) \neq A^{2}-B^{2}, \text { where } A^{2}=A A \text { and } B^{2}=B B $$

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{l} x^{2}-y^{2}=4 \\ x^{2}+y^{2}=12 \end{array}\right. $$

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{x^{5}-5 x^{4}+7 x^{3}-x^{2}-4 x+12}{x^{3}-3 x^{2}} $$

If $$ A=\left[\begin{array}{llll} a & b & 0 & 0 \\ c & d & 0 & 0 \\ 0 & 0 & e & f \\ 0 & 0 & g & h \end{array}\right] $$ show that $$ |A|=\left|\begin{array}{ll} a & b \\ c & d \end{array}\right|\left|\begin{array}{ll} e & f \\ g & h \end{array}\right| . $$

\(\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|+\left|\begin{array}{ll}a & b \\ e & f\end{array}\right|=\left|\begin{array}{cc}a & b \\ c+e & d+f\end{array}\right|\)

Let \(A=\left[\begin{array}{rr}1 & 2 \\ 0 & -3\end{array}\right], \quad B=\left[\begin{array}{rr}2 & -1 \\ 3 & 1\end{array}\right], \quad C=\left[\begin{array}{rr}3 & 1 \\ -2 & 0\end{array}\right]\) $$ (A+B)(A+B) \neq A^{2}+2 A B+B^{2} $$

Particle acceleration If a particle moves along a coordinate line with a constant acceleration \(a\) (in cm \(/ \mathrm{sec}^{2}\) ), then at time \(t\) (in seconds) its distance \(s(t)\) (in centimeters) from the origin is $$ s(t)=\frac{1}{2} a t^{2}+v_{0} t+s_{0} $$ for velocity \(v_{0}\) and distance \(s_{0}\) from the origin at \(t=0\). If the distances of the particle from the origin at \(t=\frac{1}{2}, t=1\), and \(t=\frac{3}{2}\) are 7,11 , and 17 , respectively, find \(a, v_{0}\), and \(s_{0}\).

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{l} 6 x^{3}-y^{3}=1 \\ 3 x^{3}+4 y^{3}=5 \end{array}\right. $$

A woman has $$\$ 19,000$$ to invest in two funds that pay simple interest at the rates of \(4 \%\) and \(6 \%\) per year. Interest on the \(4 \%\) fund is tax- exempt; however, income tax must be paid on interest on the \(6 \%\) fund. Being in a high tax bracket, the woman does not wish to invest the entire sum in the \(6 \%\) account. Is there a way of investing the money so that she will receive \(\$ 1000\) in interest at the end of one year?

If \(A=\left(a_{i j}\right)\) and \(B=\left(b_{i j}\right)\) are arbitrary square matrices of order 2 , show that \(|A B|=|A||B|\).

Electrical currents Shown in the figure is a schematic of an electrical circuit containing three resistors, a 6-volt battery, and a 12 -volt battery. It can be shown, using Kirchhoff's laws, that the three currents \(I_{1}, I_{2}\), and \(I_{3}\) are solutions of the following system of equations: $$ \left\\{\begin{array}{rlr} I_{1}-I_{2}+I_{3} & =0 \\ R_{1} I_{1}+R_{2} I_{2} & =6 \\ R_{2} I_{2}+R_{3} I_{3} & =12 \end{array}\right. $$ Find the three currents if (a) \(R_{1}=R_{2}=R_{3}=3\) ohms (b) \(R_{1}=4\) ohms, \(R_{2}=1 \mathrm{ohm}\), and \(R_{3}=4 \mathrm{ohms}\) Exercise 29

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{rr} x+2 y-z= & -1 \\ 2 x-y+z= & 9 \\ x+3 y+3 z= & 6 \end{array}\right. $$

A bobcat population is classified by age into kittens (less than 1 year old) and adults (at least 1 year old). All adult females, including those born the prior year, have a litter each June, with an average litter size of 3 kittens. The springtime population of bobcats in a certain area is estimated to be 6000 , and the male-female ratio is one. Estimate the number of adults and kittens in the population.