Chapter 5

The length of one side of a triangle is \(a \mathrm{ft}\). The other sides of the triangle are \(4 \mathrm{ft}\) longer and \(3 \mathrm{ft}\) shorter than this side. a. If \(a=\) length of one side, write polynomial expressions in \(a\) that represent the lengths of the other sides, and draw a diagram of the triangle. Do not include the units. b. Write a polynomial expression in \(a\) that represents the perimeter.

\(9.3 \times 10^{3} \mathrm{~kg}+4.8 \times 10^{4} \mathrm{~kg}\)

\(\left(\frac{2 p}{3 q}\right)^{2}(4 p)^{2}\)

Problem: Use long division to simplify \(\left(x^{2}-8 x+12\right) \div(x-4)\) Incorrect Answer: $$ \begin{array}{r} x - 4 \longdiv { x ^ { 2 } - 8 x + 1 2 } \\ \frac{-\left(x^{2}-4 x\right)}{-12 x+12} \\ \frac{-(-12 x+48)}{-36} \end{array} $$

The length of one side of a triangle is \(a \mathrm{ft}\). The other sides of the triangle are \(6 \mathrm{ft}\) longer and \(9 \mathrm{ft}\) shorter than this side. a. If \(a=\) length of one side, write polynomial expressions in \(a\) that represent the lengths of the other sides, and draw a diagram of the triangle. Do not include the units. b. Write a polynomial expression in \(a\) that represents the perimeter.

\(7.9 \times 10^{4} \mathrm{~km}+6.8 \times 10^{5} \mathrm{~km}\)

\(\left(\frac{3 a}{5 b}\right)^{2}(2 a)^{2}\)

Problem: Use long division to simplify $$ \left(2 x^{2}+5 x+3\right) \div(x-8) $$ Incorrect Answer: $$ \begin{array}{r} x - 8 \longdiv { 2 x ^ { 2 } + 5 x + 3 } \\ \frac{-\left(x^{2}-8 x\right)}{13 x+3} \\ \frac{-(13 x-104)}{107} \end{array} $$

The height of a triangle is \(6 \mathrm{~cm}\) less than the base. a. If \(b=\) base, write a polynomial expression in \(b\) that represents the height, and draw a diagram of the triangle. Do not include the units. b. Write a polynomial expression in \(b\) that represents the area.

\(x\left(\frac{5}{6} x-8\right)+2\left(\frac{3}{4} x-9\right)\)

\(C=\frac{7.7 \times 10^{-13}}{5 \times 10^{-6}}\)

\(6 z\left(z^{2}\right)+(6 z)(8 z)+6 z(4)\)

Problem: Simplify: \(\left(6 x^{2}+8 x-3\right) \div(3 x)\) Incorrect Answer: \(\left(6 x^{2}+8 x-3\right) \div(3 x)\) $$ \begin{aligned} &=\frac{6 x^{2}}{3 x}+\frac{8 x}{3 x}-\frac{3}{3 x} \\ &=2 x+\frac{8}{3}-x \\ &=x+\frac{8}{3} \end{aligned} $$

\(m\left(\frac{3}{4} m-2\right)+7\left(\frac{1}{3} m-5\right)\)

\(C=\frac{5 \times 10^{-8}}{2 \times 10^{-2}}\)

\(9 a\left(8 a^{2}\right)+9 a(4 a)+9 a(6)\)

The base of a triangle is 4 in. longer than twice the height. a. If \(h=\) height, write a polynomial expression in \(h\) that represents the base, and draw a diagram of the triangle. Do not include the units. b. Write a polynomial expression in \(h\) that represents the area.

\(n=\frac{\left(1 \times 10^{2}\right)\left(2 \times 10^{-4}\right)}{(8.31)\left(2.93 \times 10^{2}\right)}\)

Simplify: \(\left(\frac{x^{9}}{x^{3}}\right)^{2}\) a. Use the quotient rule first. Then use the power rule. b. Use the many bases rule for a quotient first. Next, use the power rule. Finally, use the quotient rule. c. Compare the simplified expressions from part a and part b.

\(m(9 m-2)-7(4 m-5)\)

\(F=\frac{\left(7.5 \times 10^{1}\right)\left(2.7 \times 10^{1}\right)}{1 \times 10^{-2}}\)

Simplify: \(\left(\frac{z^{7}}{z^{4}}\right)^{2}\) a. Use the quotient rule first. Then use the power rule. b. Use the many bases rule for a quotient first. Next, use the power rule. Finally, use the quotient rule. c. Compare the simplified expressions from part a and part b.

Describe the purpose of the vertical line test.

A square is a rectangle in which the lengths of all four sides are equal. If \(s=\) length of a side, write a polynomial expression in \(s\) that represents the perimeter.

85\. \(10 c^{2}(c-9)+c\left(c^{2}+12\right)\)

\(M=\frac{(4)(3.14)\left(1 \times 10^{-7}\right)(6)}{2(3.14)\left(5 \times 10^{-3}\right)}\)

Write a linear equation that represents the relationship between the number of hours to do an inspection, \(x\), and the total cost of the inspection, \(y\). In North Dakota, which has about 200 potato farms, GAP inspections are handled by the state Seed Department. Ken Bertsch, the state seed commissioner, said there is a flat fee of \(\$ 50\) besides charges of \(\$ 75\) per hour. (Source: Times-News, www.magicvalley.com, Oct. 1, 2007)

Is the set \(\\{(J o h n, A),(M a r y, A)\), (Miguel, A), (Lee, A)\\} a function? Explain.

A square is a rectangle in which the lengths of all four sides are equal. If \(s=\) length of a side, write a polynomial expression in \(s\) that represents the area.

\(L=\frac{9.45 \times 10^{6}}{\left(5 \times 10^{1}\right)(9.8)(2)}\)

Find last year's (2010) enrollment. The district's 2011 head count, released Friday, shows this year's enrollment is about 1,500 students higher than last-a 3 percent increase. (Source: www.nwsource.com, Oct. 16, 2011)

A drafter is making enlargements of a rectangular drawing that preserve the relative width and length of the drawing. The length of the drawing is five- fourths of the width. a. If \(W=\) width, write a polynomial expression in \(W\) that represents the length, and draw a diagram of the rectangle. Do not include the units. b. Write a polynomial expression in \(W\) that represents the perimeter. c. Write a polynomial expression in \(W\) that represents the area.

A periodontist removes a rectangular piece from the roof of a patient's mouth to use as a gum graft. The graft is \(1.6 \mathrm{~cm}\) long, \(9 \mathrm{~mm}\) wide, and \(2 \mathrm{~mm}\) thick. Find the volume of the graft in cubic millimeters. \((1 \mathrm{~cm}=10 \mathrm{~mm}\).) (Source: Bradley Morlock, DDS)

\(f=\frac{\left(1.34 \times 10^{1}\right)^{2}}{\left(5 \times 10^{1}\right)(9.8)}\)

A community has 37,500 low-wage workers. Predict how many do not receive paid sick days. Three out of four low-wage workers in the private sector do not have employer- provided health insurance, while eight out of nine do not participate in a pension plan. Three-fourths of low-wage workers do not receive paid sick days, so if they need to miss two days work because they are sick or their child is sick, they receive no pay for those days and of ten risk getting fired. (Source: Greenhouse, Steven, The Big Squeeze: Tough Times for the American Worker, Knopf, 2008)

A rectangular parking lot is three times as wide as it is long. a. If \(L=\) length, write a polynomial expression in \(L\) that represents the width, and draw a diagram of the rectangle. Do not include the units. b. Write a polynomial expression in \(L\) that represents the perimeter. c. Write a polynomial expression in \(L\) that represents the area.

Find the number of Haitians who live in poverty. Haiti occupies an area roughly the size of Maryland on the Caribbean island of Hispaniola, which it shares with the Dominican Republic. Nearly all of the \(8.7\) million residents are of African descent and speak Creole and French. The capital is Port-au-Prince. The country is the poorest in the Western Hemisphere, with four out of five people living in poverty and more than half in abject poverty. (Source: www nytimes.com, Dec. 27, 2011)

\(v=\frac{5 \times 10^{4}}{\left(4 \times 10^{3}\right)\left(7.99 \times 10^{-1}\right)}\)

A rectangular parking lot is three times as long as it is wide. a. If \(W=\) width, write a polynomial expression in \(W\) that represents the length, and draw a diagram of the rectangle. Do not include the units. b. Write a polynomial expression in \(W\) that represents the perimeter. c. Write a polynomial expression in \(W\) that represents the area.

The board designated unrestricted endowment at the Walker Art Center in Minneapolis increased from \(\$ 22,140,129\) in 2009 to \(\$ 27,590,778\) in 2010 . Find the percent increase. Round to the nearest percent. (Source: www.walkerart.org)

\(d=\frac{\left(3.5 \times 10^{5}\right)}{\left(1 \times 10^{3}\right)(9.8)(5.7)}\)

\((-7)^{5}\)

Concentrated sulfuric acid is \(98 \%\) sulfuric acid. The liquid mixture in lead batteries is \(33.5 \%\) sulfuric acid. Use a system of linear equations to find the amount of concentrated sulfuric acid and the amount of water needed to make \(5 \mathrm{~L}\) of the liquid mixture for a lead battery. Round to the nearest tenth.

Problem: Simplify: \(\left(6 x^{2}+8 x-4\right)-\left(3 x^{2}+5 x-1\right)\) Incorrect Answer: \(\left(6 x^{2}+8 x-4\right)-\left(3 x^{2}+5 x-1\right)\) $$ \begin{aligned} &=6 x^{2}-3 x^{2}+8 x-5 x-4-1 \\ &=3 x^{2}+3 x-5 \end{aligned} $$

\(h=\frac{2\left(7.3 \times 10^{-2}\right)}{\left(1 \times 10^{3}\right)(9.8)\left(6 \times 10^{-5}\right)}\)

\((-5)^{11}\)

Problem: Simplify: \(3 x(x-4)+8 x(9 x-3)\) $$ \text { Incorrect Answer: } \begin{aligned} & 3 x(x-4)+8 x(9 x-3) \\ &=3 x^{2}-12 x+72 x^{2}-3 \\ &=75 x^{2}-12 x-3 \end{aligned} $$

\(r=\frac{(3.14)\left(3 \times 10^{-4}\right)^{4}\left(5.15 \times 10^{3}\right)}{8\left(2.7 \times 10^{-3}\right)\left(2.0 \times 10^{-2}\right)}\)

Problem: Simplify: \(\left(5 x^{2}-8\right)-(3 x-9)\) Incorrect Answer: \(\left(5 x^{2}-8\right)-(3 x-9)\) $$ \begin{aligned} &=5 x^{2}-8-3 x+9 \\ &=2 x^{2}+1 \end{aligned} $$

\(v=\frac{\left(2.6 \times 10^{-3}\right)\left(3 \times 10^{3}\right)}{\left(1.05 \times 10^{3}\right)\left(2 \times 10^{-3}\right)}\)