7.E: Rational Expressions(Exercises)

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7.1: Negative Exponents

In Exercises 1-8, simplify the given expression.

1) $\left(\dfrac{1}{7}\right)^{-1}$

Answer: $7$

2) $\left(-\dfrac{3}{5}\right)^{-1}$

3) $\left(-\dfrac{8}{9}\right)^{-1}$

Answer: $-\dfrac{9}{8}$

4) $\left(-\dfrac{3}{2}\right)^{-1}$

5) $(18)^{-1}$

Answer: $\dfrac{1}{18}$

6) $(-11)^{-1}$

7) $(16)^{-1}$

Answer: $\dfrac{1}{16}$

8) $(7)^{-1}$

In Exercises 9-16, simplify the given expression.

9) $a^{-9} a^{3}$

Answer: $a^{-6}$

10) $x^{-5} x^{-5}$

11) $b^{-9} b^{8}$

Answer: $b^{-1}$

12) $v^{-7} v^{-2}$

13) $2^{9} \cdot 2^{-4}$

Answer: $2^{5}$

14) $2^{2} \cdot 2^{-7}$

15) $9^{-6} \cdot 9^{-5}$

Answer: $9^{-11}$

16) $9^{7} \cdot 9^{-5}$

In Exercises 17-24, simplify the given expression.

17) $\dfrac{2^{6}}{2^{-8}}$

Answer: $2^{14}$

18) $\dfrac{6^{8}}{6^{-1}}$

19) $\dfrac{z^{-1}}{z^{9}}$

Answer: $z^{-10}$

20) $\dfrac{w^{-4}}{w^{3}}$

21) $\dfrac{w^{-9}}{w^{7}}$

Answer: $w^{-16}$

22) $\dfrac{r^{5}}{r^{-1}}$

23) $\dfrac{7^{-3}}{7^{-1}}$

Answer: $7^{-2}$

24) $\dfrac{6^{-8}}{6^{6}}$

In Exercises 25-32, simplify the given expression.

25) $\left(t^{-1}\right)^{4}$

Answer: $t^{-4}$

26) $\left(a^{8}\right)^{-7}$

27) $\left(6^{-6}\right)^{7}$

Answer: $6^{-42}$

28) $\left(2^{-7}\right)^{-7}$

29) $\left(z^{-9}\right)^{-9}$

Answer: $z^{81}$

30) $\left(c^{6}\right)^{-2}$

31) $\left(3^{-2}\right)^{3}$

Answer: $3^{-6}$

32) $\left(8^{-1}\right)^{6}$

In Exercises 33-40, simplify the given expression.

33) $4^{-3}$

Answer: $\dfrac{1}{64}$

34) $5^{-2}$

35) $2^{-4}$

Answer: $\dfrac{1}{16}$

36) $(-3)^{-4}$

37) $\left(\dfrac{1}{2}\right)^{-5}$

Answer: $32$

38) $\left(\dfrac{1}{3}\right)^{-3}$

39) $\left(-\dfrac{1}{2}\right)^{-5}$

Answer: $-32$

40) $\left(\dfrac{1}{2}\right)^{-4}$

In Exercises 41-56, simplify the given expression.

41) $\left(4 u^{-6} v^{-9}\right)\left(5 u^{8} v^{-8}\right)$

Answer: $20 u^{2} v^{-17}$

42) $\left(6 a^{-9} c^{-6}\right)\left(-8 a^{8} c^{5}\right)$

43) $\left(6 x^{-6} y^{-5}\right)\left(-4 x^{4} y^{-2}\right)$

Answer: $-24 x^{-2} y^{-7}$

44) $\left(5 v^{-3} w^{-8}\right)\left(8 v^{-9} w^{5}\right)$

45) $\dfrac{-6 x^{7} z^{9}}{4 x^{-9} z^{-2}}$

Answer: $-\dfrac{3}{2} x^{16} z^{11}$

46) $\dfrac{2 u^{-2} v^{6}}{6 u^{2} v^{-1}}$

47) $\dfrac{-6 a^{9} c^{6}}{-4 a^{-5} c^{-7}}$

Answer: $\dfrac{3}{2} a^{14} c^{13}$

48) $\dfrac{-4 u^{-4} w^{4}}{8 u^{-8} w^{-7}}$

49) $\left(2 v^{-2} w^{4}\right)^{-5}$

Answer: $\dfrac{1}{32} v^{10} w^{-20}$

50) $\left(3 s^{-6} t^{5}\right)^{-4}$

51) $\left(3 x^{-1} y^{7}\right)^{4}$

Answer: $81 x^{-4} y^{28}$

52) $\left(-4 b^{-8} c^{-4}\right)^{3}$

53) $\left(2 x^{6} z^{-7}\right)^{5}$

Answer: $32 x^{30} z^{-35}$

54) $\left(-4 v^{4} w^{-9}\right)^{3}$

55) $\left(2 a^{-4} c^{8}\right)^{-4}$

Answer: $\dfrac{1}{16} a^{16} c^{-32}$

56) $\left(11 b^{9} c^{-1}\right)^{-2}$

In Exercises 57-76, clear all negative exponents from the given expression.

57) $\dfrac{x^{5} y^{-2}}{z^{3}}$

Answer: $\dfrac{x^{5}}{y^{2} z^{3}}$

58) $\dfrac{x^{4} y^{-9}}{z^{7}}$

59) $\dfrac{r^{9} s^{-2}}{t^{3}}$

Answer: $\dfrac{r^{9}}{s^{2} t^{3}}$

60) $\dfrac{u^{5} v^{-3}}{w^{2}}$

61) $\dfrac{x^{3}}{y^{-8} z^{5}}$

Answer: $\dfrac{x^{3} y^{8}}{z^{5}}$

62) $\dfrac{x^{9}}{y^{-4} z^{3}}$

63) $\dfrac{u^{9}}{v^{-4} w^{7}}$

Answer: $\dfrac{u^{9} v^{4}}{w^{7}}$

64) $\dfrac{a^{7}}{b^{-8} c^{6}}$

65) $\left(7 x^{-1}\right)\left(-7 x^{-1}\right)$

Answer: $\dfrac{-49}{x^{2}}$

66) $\left(3 a^{-8}\right)\left(-7 a^{-7}\right)$

67) $\left(8 a^{-8}\right)\left(7 a^{-7}\right)$

Answer: $\dfrac{56}{a^{15}}$

68) $\left(-7 u^{3}\right)\left(-8 u^{-6}\right)$

69) $\dfrac{4 x^{-9}}{8 x^{3}}$

Answer: $\dfrac{1}{2 x^{12}}$

70) $\dfrac{2 t^{-8}}{-6 t^{9}}$

71) $\dfrac{6 c^{2}}{-4 c^{7}}$

Answer: $-\dfrac{3}{2 c^{5}}$

72) $\dfrac{6 v^{-9}}{-8 v^{-4}}$

73) $\left(-3 s^{9}\right)^{-4}$

Answer: $\dfrac{1}{81 s^{36}}$

74) $\left(-3 s^{8}\right)^{-4}$

75) $\left(2 y^{4}\right)^{-5}$

Answer: $\dfrac{1}{32 y^{20}}$

76) $\left(2 w^{4}\right)^{-5}$

7.2: Scientiﬁc Notation

In Exercises 1-8, write each of the following in decimal format.

1) $10^{-4}$

Answer: $0.0001$

2) $10^{-13}$

3) $10^{-8}$

Answer: $0.00000001$

4) $10^{-9}$

5) $10^{8}$

Answer: $100,000,000$

6) $10^{14}$

7) $10^{7}$

Answer: $10,000,000$

8) $10^{9}$

In Exercises 9-16, write each of the following in decimal format.

9) $6506399.9 \times 10^{-4}$

Answer: $650.63999$

10) $19548.4 \times 10^{-2}$

11) $3959.430928 \times 10^{2}$

Answer: $395943.0928$

12) $976.841866 \times 10^{2}$

13) $440906.28 \times 10^{-4}$

Answer: $44.090628$

14) $9147437.4 \times 10^{-4}$

15) $849.855115 \times 10^{4}$

Answer: $8498551.15 $

16) $492.4414 \times 10^{3}$

In Exercises 17-24, convert each of the given numbers into scientiﬁc notation.

17) $390000$

Answer: $3.9 \times 10^{5}$

18) $0.0004902$

19) $0.202$

Answer: $2.02 \times 10^{-1}$

20) $3231$

21) $0.81$

Answer: $8.1 \times 10^{-1}$

22) $83400$

23) $0.0007264$

Answer: $7.264 \times 10^{-4}$

24) $0.00395$

In Exercises 25-32, convert each of the given expressions into scientiﬁc notation.

25) $0.04264 \times 10^{-4}$

Answer: $4.264 \times 10^{-6}$

26) $0.0019 \times 10^{-1}$

27) $130000 \times 10^{3}$

Answer: $1.3 \times 10^{8}$

28) $738 \times 10^{-1}$

29) $30.04 \times 10^{5}$

Answer: $3.004 \times 10^{6}$

30) $76000 \times 10^{-1}$

31) $0.011 \times 10^{1}$

Answer: $1.1 \times 10^{-1}$

32) $496000 \times 10^{-3}$

In Exercises 33-38, each of the following numbers are examples of numbers reported on the graphing calculator in scientiﬁc notation. Express each in plain decimal notation.

33) $1.134 \mathrm{E}-1$

Answer: $0.1134$

34) $1.370 \mathrm{E}-4$

35) $1.556 \mathrm{E}-2$

Answer: $0.01556$

36) $1.802 \mathrm{E} 4$

37) $1.748 \mathrm{E}-4$

Answer: $0.0001748$

38) $1.402 \mathrm{E} 0$

In Exercises 39-42, ﬁrst, use the technique of Example 7.2.11 to approximate the given product without the use of a calculator. Next, use the MODE button to set you calculator in SCI and FLOAT mode, then enter the given product using scientiﬁc notation. When reporting your answer, report all digits shown in your calculator view screen.

39) $\left(2.5 \times 10^{-1}\right)\left(1.6 \times 10^{-7}\right)$

Answer: $4 \times 10^{-8}$

40) $\left(2.91 \times 10^{-1}\right)\left(2.81 \times 10^{-4}\right)$

41) $\left(1.4 \times 10^{7}\right)\left(1.8 \times 10^{-4}\right)$

Answer: $2.52 \times 10^{3}$

42) $\left(7.48 \times 10^{7}\right)\left(1.19 \times 10^{6}\right)$

In Exercises 43-46, ﬁrst, use the technique of Example 7.2.12 to approximate the given quotient without the use of a calculator. Next, push the MODE button, then highlight SCI mode and press ENTER. Move your cursor to the same row containing the FLOAT command, then highlight the number $2$ and press ENTER. This will round your answers to two decimal places. Press 2nd MODE to quit the MODE menu. With these settings, enter the given expression using scientiﬁc notation. When entering your answer, report all digits shown in the viewing window.

43) $\dfrac{3.2 \times 10^{-5}}{2.5 \times 10^{-7}}$

Answer: $1.28 \times 10^{2}$

44) $\dfrac{6.47 \times 10^{-5}}{1.79 \times 10^{8}}$

45) $\dfrac{5.9 \times 10^{3}}{2.3 \times 10^{5}}$

Answer: $2.57 \times 10^{-2}$

46) $\dfrac{8.81 \times 10^{-9}}{3.06 \times 10^{-1}}$

47) Overall the combined weight of biological material – animals, plants, insects, crops, bacteria, and so on – has been estimated to be at about $75$ billion tons or $6.8×10^{13}$ kg (https://en.Wikipedia.org/wiki/Nature). If the Earth has mass of $5.9736×10^{24}$ kg, what is the percent of the Earth’s mass that is made up of biomass?

Answer: $1.14 \times 10^{-11}$

48) The Guinness World Record for the longest handmade noodle was set on March 20, 2011. The $1,704$-meter-long stretch of noodle was displayed during a noodle-making activity at a square in Southwest China’s Yunnan province. Meigan estimates that the average width of the noodle (it’s diameter) to be the same as her index ﬁnger or $1.5$ cm. Using the volume formula for a cylinder ($V = \pi r^2h$) estimate the volume of the noodle in cubic centimeters

49) Assume there are $1.43×10^6$ miles of paved road in the United States. If you could travel at an average of $65$ miles per hour nonstop, how many days would it take you to travel over all of the paved roads in the USA? How many years?

Answer: $916.7$ days, $2.5$ yr

50) The population of the USA in mid-2011 was estimated to be $3.12×10^8$ people and the world population at that time to be about $7.012×10^9$ people. What percentage of the world population live in the USA?

7.3: Simplifying Rational Expressions

In Exercises 1-8, simplify each of the given experssions.

1) $\dfrac{12}{s^{2}} \cdot \dfrac{s^{5}}{9}$

Answer: $\dfrac{4 s^{3}}{3}$

2) $\dfrac{6}{x^{4}} \cdot \dfrac{x^{2}}{10}$

3) $\dfrac{12}{v^{3}} \cdot \dfrac{v^{4}}{10}$

Answer: $\dfrac{6 v}{5}$

4) $\dfrac{10}{t^{4}} \cdot \dfrac{t^{5}}{12}$

5) $\dfrac{s^{5}}{t^{4}} \div \dfrac{9 s^{2}}{t^{2}}$

Answer: $\dfrac{s^{3}}{9 t^{2}}$

6) $\dfrac{s^{2}}{t^{2}} \div \dfrac{6 s^{4}}{t^{4}}$

7) $\dfrac{b^{4}}{c^{4}} \div \dfrac{9 b^{2}}{c^{2}}$

Answer: $\dfrac{b^{2}}{9 c^{2}}$

8) $\dfrac{b^{5}}{c^{4}} \div \dfrac{8 b^{2}}{c^{2}}$

In Exercises 9-14, simplify each of the given expressions.

9) $-\dfrac{10 s}{18}+\dfrac{19 g}{18}$

Answer: $\dfrac{s}{2}$

10) $-\dfrac{14 y}{2}+\dfrac{10 y}{2}$

11) $\dfrac{5}{9 c}-\dfrac{17}{9 c}$

Answer: $-\dfrac{4}{3 c}$

12) $\dfrac{19}{14 r}-\dfrac{17}{14 r}$

13) $-\dfrac{8 x}{15 y z}-\dfrac{16 x}{15 y z}$

Answer: $-\dfrac{8 x}{5 y z}$

14) $-\dfrac{17 a}{20 b c}-\dfrac{9 a}{20 b c}$

In Exercises 15-20, simplify each of the given expressions.

15) $\dfrac{9 z}{10}+\dfrac{5 z}{2}$

Answer: $\dfrac{17 z}{5}$

16) $\dfrac{7 u}{2}+\dfrac{11 u}{6}$

17) $\dfrac{3}{10 v}-\dfrac{4}{5 v}$

Answer: $-\dfrac{1}{2 v}$

18) $\dfrac{9}{10 v}-\dfrac{7}{2 v}$

19) $-\dfrac{8 r}{5 s t}-\dfrac{9 r}{10 s t}$

Answer: $-\dfrac{5 r}{2 s t}$

20) $-\dfrac{7 x}{6 y z}-\dfrac{3 x}{2 y z}$

In Exercises 21-32, simplify each of the given expressions.

21) $\dfrac{11}{18 r s^{2}}+\dfrac{5}{24 r^{2} s}$

Answer: $\dfrac{44 r+15 s}{72 r^{2} s^{2}}$

22) $\dfrac{5}{12 u w^{2}}+\dfrac{13}{54 u^{2} w}$

23) $\dfrac{5}{24 r s^{2}}+\dfrac{17}{36 r^{2} s}$

Answer: $\dfrac{15 r+34 s}{72 r^{2} s^{2}}$

24) $\dfrac{13}{54 v w^{2}}+\dfrac{19}{24 v^{2} w}$

25) $\dfrac{7}{36 y^{3}}+\dfrac{11}{48 z^{3}}$

Answer: $\dfrac{28 z^{3}+33 y^{3}}{144 y^{3} z^{3}}$

26) $\dfrac{19}{36 x^{3}}+\dfrac{5}{48 y^{3}}$

27) $\dfrac{5}{48 v^{3}}+\dfrac{13}{36 u v^{3}}$

Answer: $\dfrac{15 w^{3}+52 v^{3}}{144 v^{3} w^{3}}$

28) $\dfrac{7}{72 r^{3}}+\dfrac{17}{48 s^{3}}$

29) $\dfrac{11}{50 x y}-\dfrac{9}{40 y z}$

Answer: $\dfrac{44 z-45 x}{200 x y z}$

30) $\dfrac{9}{50 r s}-\dfrac{13}{40 s t}$

31) $\dfrac{19}{50 a b}-\dfrac{17}{40 b c}$

Answer: $\dfrac{76 c-85 a}{200 a b c}$

32) $\dfrac{9}{50 r s}-\dfrac{11}{40 s t}$

In Exercises 33-48, use the distributive property to divide each term in the numerator by the term in the denominator.

33) $\dfrac{6 v+12}{3}$

Answer: $2 v+4$

34) $\dfrac{28 u+36}{4}$

35) $\dfrac{25 u+45}{5}$

Answer: $5 u+9$

36) $\dfrac{16 x+4}{2}$

37) $\dfrac{2 s-4}{s}$

Answer: $2-\dfrac{4}{g}$

38) $\dfrac{7 r-8}{r}$

39) $\dfrac{3 r-5}{r}$

Answer: $3-\dfrac{5}{r}$

40) $\dfrac{4 u-2}{u}$

41) $\dfrac{3 x^{2}-8 x-9}{x^{2}}$

Answer: $3-\dfrac{8}{x}-\dfrac{9}{x^{2}}$

42) $\dfrac{4 b^{2}-5 b-8}{b^{2}}$

43) $\dfrac{2 x^{2}-3 x-6}{x^{2}}$

Answer: $2-\dfrac{3}{x}-\dfrac{6}{x^{2}}$

44) $\dfrac{6 u^{2}-5 u-2}{u^{2}}$

45) $\dfrac{12 t^{2}+2 t-16}{12 t^{2}}$

Answer: $1+\dfrac{1}{6 t}-\dfrac{4}{3 t^{2}}$

46) $\dfrac{18 b^{2}+9 b-15}{18 b^{2}}$

47) $\dfrac{4 s^{2}+2 s-10}{4 s^{2}}$

Answer: $1+\dfrac{1}{2 s}-\dfrac{5}{2 s^{2}}$

48) $\dfrac{10 w^{2}+12 w-2}{10 w^{2}}$

7.4: Solving Rational Equations

In Exercises 1-8, solve the equation.

1) $x=11+\dfrac{26}{x}$

Answer: $-2,13$

2) $x=7+\dfrac{60}{x}$

3) $1-\dfrac{12}{x}=-\dfrac{27}{x^{2}}$

Answer: $3,9$

4) $1+\dfrac{6}{x}=\dfrac{7}{x^{2}}$

5) $1-\dfrac{10}{x}=\dfrac{11}{x^{2}}$

Answer: $11,-1$

6) $1-\dfrac{20}{x}=-\dfrac{96}{x^{2}}$

7) $x=7+\dfrac{44}{x}$

Answer: $-4,11$

8) $x=2+\dfrac{99}{x}$

In Exercises 9-16, solve the equation.

9) $12 x=97-\dfrac{8}{x}$

Answer: $8, \dfrac{1}{12}$

10) $7 x=-19-\dfrac{10}{x}$

11) $20+\dfrac{19}{x}=-\dfrac{3}{x^{2}}$

Answer: $-\dfrac{3}{4},-\dfrac{1}{5}$

12) $33-\dfrac{8}{x}=\dfrac{1}{x^{2}}$

13) $8 x=19-\dfrac{11}{x}$

Answer: $\dfrac{11}{8},1$

14) $28 x=25-\dfrac{3}{x}$

15) $40+\dfrac{6}{x}=\dfrac{1}{x^{2}}$

Answer: $-\dfrac{1}{4}, \dfrac{1}{10}$

16) $18+\dfrac{11}{x}=-\dfrac{1}{x^{2}}$

In Exercises 17-20, solve each equation algebraically, then use the calculator to check your solutions.

17) $36 x=-13-\dfrac{1}{x}$

Answer: $-\dfrac{1}{9},-\dfrac{1}{4}$

18) $9 x=43+\dfrac{10}{x}$

19) $14 x=9-\dfrac{1}{x}$

Answer: $\dfrac{1}{2}, \dfrac{1}{7}$

20) $3 x=16-\dfrac{20}{x}$

In Exercises 21-24, solve the equation algebraically, then solve the equation using the graphing calculator using the technique shown in Example 7.4.3. Report your solution using the Calculator Submission Guidelines demonstrated in Example 7.4.3.

21) $1-\dfrac{1}{x}=\dfrac{12}{x^{2}}$

Answer: $-3,4$

22) $1+\dfrac{11}{x}=-\dfrac{28}{x^{2}}$

23) $2 x=3+\dfrac{44}{x}$

Answer: $-4, \dfrac{11}{2}$

24) $2 x=9-\dfrac{4}{x}$

25) The sum of a number and its reciprocal is $\dfrac{5}{2}$. Find the number.

Answer: $2, \dfrac{1}{2}$

26) The sum of a number and its reciprocal is $\dfrac{65}{8}$. Find the number.

27) The sum of a number and 8 times its reciprocal is $\dfrac{17}{3}$. Find all possible solutions.

Answer: $3, \dfrac{8}{3}$

28) The sum of a number and 4 times its reciprocal is $\dfrac{17}{2}$. Find all possible solutions.

7.5: Direct and Inverse Variation

1) Given that $s$ is proportional to $t$ and the fact that $s = 632$ when $t = 79$, determine the value of $s$ when $t = 50$.

Answer: $400$

2) Given that $s$ is proportional to $t$ and the fact that $s = 264$ when $t = 66$, determine the value of $s$ when $t = 60$.

3) Given that $s$ is proportional to the cube of $t$ and the fact that $s = 1588867$ when $t = 61$, determine the value of $s$ when $t = 63$.

Answer: $1750329$

4) Given that $d$ is proportional to the cube of $t$ and the fact that $d = 318028$ when $t = 43$, determine the value of $d$ when $t = 76$.

5) Given that $q$ is proportional to the square of $c$ and the fact that $q = 13448$ when $c = 82$, determine the value of $q$ when $c = 29$.

Answer: $1682$

6) Given that $q$ is proportional to the square of $c$ and the fact that $q = 3125$ when $c = 25$, determine the value of $q$ when $c = 87$.

7) Given that $y$ is proportional to the square of $x$ and the fact that $y = 14700$ when $x = 70$, determine the value of $y$ when $x = 45$.

Answer: $6075$

8) Given that $y$ is proportional to the square of $x$ and the fact that $y = 2028$ when $x = 26$, determine the value of $y$ when $x = 79$.

9) Given that $F$ is proportional to the cube of $x$ and the fact that $F = 214375$ when $x = 35$, determine the value of $F$ when $x = 36$.

Answer: $233280$

10) Given that $d$ is proportional to the cube of $t$ and the fact that $d = 2465195$ when $t = 79$, determine the value of $d$ when $t = 45$.

11) Given that $d$ is proportional to $t$ and the fact that $d = 496$ when $t = 62$, determine the value of $d$ when $t = 60$.

Answer: $480$

12) Given that $d$ is proportional to $t$ and the fact that $d = 405$ when $t = 45$, determine the value of $d$ when $t = 65$.

13) Given that $h$ is inversely proportional to $x$ and the fact that $h = 16$ when $x = 29$, determine the value of $h$ when $x = 20$.

Answer: $\dfrac {116}{5}$

14) Given that $y$ is inversely proportional to $x$ and the fact that $y = 23$ when $x = 15$, determine the value of $y$ when $x = 10$.

15) Given that $q$ is inversely proportional to the square of $c$ and the fact that $q = 11$ when $c = 9$, determine the value of $q$ when $c = 3$.

Answer: $99$

16) Given that $s$ is inversely proportional to the square of $t$ and the fact that $s = 11$ when $t = 8$, determine the value of $s$ when $t = 10$.

17) Given that $F$ is inversely proportional to $x$ and the fact that $F = 19$ when $x = 22$, determine the value of $F$ when $x = 16$.

Answer: $\dfrac {209}{8}$

18) Given that $d$ is inversely proportional to $t$ and the fact that $d = 21$ when $t = 16$, determine the value of $d$ when $t = 24$.

19) Given that $y$ is inversely proportional to the square of $x$ and the fact that $y = 14$ when $x = 4$, determine the value of $y$ when $x = 10$.

Answer: $\dfrac {56}{25}$

20) Given that $d$ is inversely proportional to the square of $t$ and the fact that $d = 21$ when $t = 8$, determine the value of $d$ when $t = 12$.

21) Given that $d$ is inversely proportional to the cube of $t$ and the fact that $d = 18$ when $t = 2$, determine the value of $d$ when $t = 3$.

Answer: $\dfrac {16}{3}$

22) Given that $q$ is inversely proportional to the cube of $c$ and the fact that $q = 10$ when $c = 5$, determine the value of $q$ when $c = 6$.

23) Given that $q$ is inversely proportional to the cube of $c$ and the fact that $q = 16$ when $c = 5$, determine the value of $q$ when $c = 6$.

Answer: $\dfrac {250}{27}$

24) Given that $q$ is inversely proportional to the cube of $c$ and the fact that $q = 15$ when $c = 6$, determine the value of $q$ when $c = 2$.

25) Joe and Mary are hanging weights on a spring in the physics lab. Each time a weight is hung, they measure the distance the spring stretches. They discover that the distance that the spring stretches is proportional to the weight hung on the spring. If a $2$ pound weight stretches the spring $16$ inches, how far will a $5$ pound weight stretch the spring?

Answer: $40$ inches

26) Liz and Denzel are hanging weights on a spring in the physics lab. Each time a weight is hung, they measure the distance the spring stretches. They discover that the distance that the spring stretches is proportional to the weight hung on the spring. If a $5$ pound weight stretches the spring $12.5$ inches, how far will a $12$ pound weight stretch the spring?

27) The intensity $I$ of light is inversely proportional to the square of the distance $d$ from the light source. If the light intensity $4$ feet from the light source is $20$ foot-candles, what is the intensity of the light $18$ feet from the light source?

Answer: $1.0$ foot-candles

28) The intensity $I$ of light is inversely proportional to the square of the distance $d$ from the light source. If the light intensity $5$ feet from the light source is $10$ foot-candles, what is the intensity of the light $10$ feet from the light source

29) Suppose that the price per person for a camping experience is inversely proportional to the number of people who sign up for the experience. If $18$ people sign up, the price per person is $\$204$. What will be the price per person if $35$ people sign up? Round your answer to the nearest dollar.

Answer: $\$105$

30) Suppose that the price per person for a camping experience is inversely proportional to the number of people who sign up for the experience. If $17$ people sign up, the price per person is $\$213$. What will be the price per person if $27$ people sign up? Round your answer to the nearest dollar.

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