Math / Algebra

QuestionAnswer the following problems. Show your solutions.
1. If $y$ varies directly as $x$ . If $y=216$ when $x=9$ , a. What is the constant of variation for the given situation? b. What is the equation of variation? c. What is $y$ when $x=12$ ? d. What is $x$ when $y=\underline{27}$ ?
2. If m varies inversely as n . If $\mathrm{m}=144$ when $\mathrm{n}=4$ , a. What is the constant of variation for the given situation? b. What is the equation of variation? c. What is the value of $m$ when $n$ is 9 ? d. What is the value of $n$ when $m$ is 12 ?
3. Suppose $a$ varies jointly with $b$ and $c$ . if $a=336, b=7$ and $c$ is 2 . a. What is the constant of variation for the given situation? b. What is the equation of variation? c. What is the value of a when $\mathrm{b}=9$ and $\mathrm{c}=3$ ? d. What is the value of $b$ when $a=720$ and $c=\underline{6}$ ? e. What is the value of $c$ when $a=960$ and $b=8$ ?
4. Suppose a varies directly as b and inversely as c . If $\mathrm{a}=12$ , when $\mathrm{b}=8$ and c is 4 . a. What is the constant of variation for the given situation? b. What is the equation of variation? c. What is the value of $a$ when $b=9$ and $c=3$ ? d. What is the value of $b$ when $a=24$ and $c=5$ ? e. What is the value of $c$ when $a=45$ and $b=15$ ?
5. If $m$ varies inversely as g and $\mathrm{m}=30$ when $\mathrm{g}=7$ , a. What is the constant of variation for the given situation? b. What is the equation of variation? c. What is m when g is 21 ? d. What is g when m is 15 ?

Studdy Solution

STEP 1

1. We are dealing with problems involving direct, inverse, and joint variation.
2. The problems require finding constants of variation, equations of variation, and solving for unknown variables.

STEP 2

1. Solve Problem 1: Direct Variation
2. Solve Problem 2: Inverse Variation
3. Solve Problem 3: Joint Variation
4. Solve Problem 4: Direct and Inverse Variation
5. Solve Problem 5: Inverse Variation

STEP 3

Solve Problem 1: Direct Variation
a. Determine the constant of variation $k$ .
Since $y$ varies directly as $x$ , we have $y = kx$ .
Given $y = 216$ when $x = 9$ , substitute these values into the equation:
$216 = k \times 9$
Solve for $k$ :
$k = \frac{216}{9} = 24$

STEP 4

b. Write the equation of variation.
The equation is $y = 24x$ .

STEP 5

c. Find $y$ when $x = 12$ .
Substitute $x = 12$ into the equation $y = 24x$ :
$y = 24 \times 12 = 288$

STEP 6

d. Find $x$ when $y = 27$ .
Substitute $y = 27$ into the equation $y = 24x$ :
$27 = 24x$
Solve for $x$ :
$x = \frac{27}{24} = \frac{9}{8}$

STEP 7

Solve Problem 2: Inverse Variation
a. Determine the constant of variation $k$ .
Since $m$ varies inversely as $n$ , we have $m = \frac{k}{n}$ .
Given $m = 144$ when $n = 4$ , substitute these values into the equation:
$144 = \frac{k}{4}$
Solve for $k$ :
$k = 144 \times 4 = 576$

STEP 8

b. Write the equation of variation.
The equation is $m = \frac{576}{n}$ .

STEP 9

c. Find $m$ when $n = 9$ .
Substitute $n = 9$ into the equation $m = \frac{576}{n}$ :
$m = \frac{576}{9} = 64$

STEP 10

d. Find $n$ when $m = 12$ .
Substitute $m = 12$ into the equation $m = \frac{576}{n}$ :
$12 = \frac{576}{n}$
Solve for $n$ :
$n = \frac{576}{12} = 48$

STEP 11

Solve Problem 3: Joint Variation
a. Determine the constant of variation $k$ .
Since $a$ varies jointly with $b$ and $c$ , we have $a = kbc$ .
Given $a = 336$ , $b = 7$ , and $c = 2$ , substitute these values into the equation:
$336 = k \times 7 \times 2$
Solve for $k$ :
$k = \frac{336}{14} = 24$

STEP 12

b. Write the equation of variation.
The equation is $a = 24bc$ .

STEP 13

c. Find $a$ when $b = 9$ and $c = 3$ .
Substitute $b = 9$ and $c = 3$ into the equation $a = 24bc$ :
$a = 24 \times 9 \times 3 = 648$

STEP 14

d. Find $b$ when $a = 720$ and $c = 6$ .
Substitute $a = 720$ and $c = 6$ into the equation $a = 24bc$ :
$720 = 24b \times 6$
Solve for $b$ :
$b = \frac{720}{144} = 5$

STEP 15

e. Find $c$ when $a = 960$ and $b = 8$ .
Substitute $a = 960$ and $b = 8$ into the equation $a = 24bc$ :
$960 = 24 \times 8 \times c$
Solve for $c$ :
$c = \frac{960}{192} = 5$

STEP 16

Solve Problem 4: Direct and Inverse Variation
a. Determine the constant of variation $k$ .
Since $a$ varies directly as $b$ and inversely as $c$ , we have $a = k \frac{b}{c}$ .
Given $a = 12$ , $b = 8$ , and $c = 4$ , substitute these values into the equation:
$12 = k \frac{8}{4}$
Solve for $k$ :
$k = \frac{12 \times 4}{8} = 6$

STEP 17

b. Write the equation of variation.
The equation is $a = 6 \frac{b}{c}$ .

STEP 18

c. Find $a$ when $b = 9$ and $c = 3$ .
Substitute $b = 9$ and $c = 3$ into the equation $a = 6 \frac{b}{c}$ :
$a = 6 \frac{9}{3} = 18$

STEP 19

d. Find $b$ when $a = 24$ and $c = 5$ .
Substitute $a = 24$ and $c = 5$ into the equation $a = 6 \frac{b}{c}$ :
$24 = 6 \frac{b}{5}$
Solve for $b$ :
$b = \frac{24 \times 5}{6} = 20$

STEP 20

e. Find $c$ when $a = 45$ and $b = 15$ .
Substitute $a = 45$ and $b = 15$ into the equation $a = 6 \frac{b}{c}$ :
$45 = 6 \frac{15}{c}$
Solve for $c$ :
$c = \frac{6 \times 15}{45} = 2$

STEP 21

Solve Problem 5: Inverse Variation
a. Determine the constant of variation $k$ .
Since $m$ varies inversely as $g$ , we have $m = \frac{k}{g}$ .
Given $m = 30$ when $g = 7$ , substitute these values into the equation:
$30 = \frac{k}{7}$
Solve for $k$ :
$k = 30 \times 7 = 210$

STEP 22

b. Write the equation of variation.
The equation is $m = \frac{210}{g}$ .

STEP 23

c. Find $m$ when $g = 21$ .
Substitute $g = 21$ into the equation $m = \frac{210}{g}$ :
$m = \frac{210}{21} = 10$

STEP 24

d. Find $g$ when $m = 15$ .
Substitute $m = 15$ into the equation $m = \frac{210}{g}$ :
$15 = \frac{210}{g}$
Solve for $g$ :
$g = \frac{210}{15} = 14$

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Studdy Solution

STEP 1

1. We are dealing with problems involving direct, inverse, and joint variation.2. The problems require finding constants of variation, equations of variation, and solving for unknown variables.

STEP 2

1. Solve Problem 1: Direct Variation2. Solve Problem 2: Inverse Variation3. Solve Problem 3: Joint Variation4. Solve Problem 4: Direct and Inverse Variation5. Solve Problem 5: Inverse Variation

STEP 3

STEP 4

b. Write the equation of variation.The equation is y=24x y = 24x y=24x.

STEP 5

c. Find y y y when x=12 x = 12 x=12.Substitute x=12 x = 12 x=12 into the equation y=24x y = 24x y=24x:y=24×12=288 y = 24 \times 12 = 288 y=24×12=288

STEP 6

d. Find x x x when y=27 y = 27 y=27.Substitute y=27 y = 27 y=27 into the equation y=24x y = 24x y=24x:27=24x 27 = 24x 27=24xSolve for x x x:x=2724=98 x = \frac{27}{24} = \frac{9}{8} x=2427​=89​

STEP 7

STEP 8

b. Write the equation of variation.The equation is m=576n m = \frac{576}{n} m=n576​.

STEP 9

c. Find m m m when n=9 n = 9 n=9.Substitute n=9 n = 9 n=9 into the equation m=576n m = \frac{576}{n} m=n576​:m=5769=64 m = \frac{576}{9} = 64 m=9576​=64

STEP 10

d. Find n n n when m=12 m = 12 m=12.Substitute m=12 m = 12 m=12 into the equation m=576n m = \frac{576}{n} m=n576​:12=576n 12 = \frac{576}{n} 12=n576​Solve for n n n:n=57612=48 n = \frac{576}{12} = 48 n=12576​=48

STEP 11

STEP 12

b. Write the equation of variation.The equation is a=24bc a = 24bc a=24bc.

STEP 13

c. Find a a a when b=9 b = 9 b=9 and c=3 c = 3 c=3.Substitute b=9 b = 9 b=9 and c=3 c = 3 c=3 into the equation a=24bc a = 24bc a=24bc:a=24×9×3=648 a = 24 \times 9 \times 3 = 648 a=24×9×3=648

STEP 14

d. Find b b b when a=720 a = 720 a=720 and c=6 c = 6 c=6.Substitute a=720 a = 720 a=720 and c=6 c = 6 c=6 into the equation a=24bc a = 24bc a=24bc:720=24b×6 720 = 24b \times 6 720=24b×6Solve for b b b:b=720144=5 b = \frac{720}{144} = 5 b=144720​=5

STEP 15

e. Find c c c when a=960 a = 960 a=960 and b=8 b = 8 b=8.Substitute a=960 a = 960 a=960 and b=8 b = 8 b=8 into the equation a=24bc a = 24bc a=24bc:960=24×8×c 960 = 24 \times 8 \times c 960=24×8×cSolve for c c c:c=960192=5 c = \frac{960}{192} = 5 c=192960​=5

STEP 16

STEP 17

b. Write the equation of variation.The equation is a=6bc a = 6 \frac{b}{c} a=6cb​.

STEP 18

c. Find a a a when b=9 b = 9 b=9 and c=3 c = 3 c=3.Substitute b=9 b = 9 b=9 and c=3 c = 3 c=3 into the equation a=6bc a = 6 \frac{b}{c} a=6cb​:a=693=18 a = 6 \frac{9}{3} = 18 a=639​=18

STEP 19

d. Find b b b when a=24 a = 24 a=24 and c=5 c = 5 c=5.Substitute a=24 a = 24 a=24 and c=5 c = 5 c=5 into the equation a=6bc a = 6 \frac{b}{c} a=6cb​:24=6b5 24 = 6 \frac{b}{5} 24=65b​Solve for b b b:b=24×56=20 b = \frac{24 \times 5}{6} = 20 b=624×5​=20

STEP 20

e. Find c c c when a=45 a = 45 a=45 and b=15 b = 15 b=15.Substitute a=45 a = 45 a=45 and b=15 b = 15 b=15 into the equation a=6bc a = 6 \frac{b}{c} a=6cb​:45=615c 45 = 6 \frac{15}{c} 45=6c15​Solve for c c c:c=6×1545=2 c = \frac{6 \times 15}{45} = 2 c=456×15​=2

STEP 21

STEP 22

b. Write the equation of variation.The equation is m=210g m = \frac{210}{g} m=g210​.

STEP 23

c. Find m m m when g=21 g = 21 g=21.Substitute g=21 g = 21 g=21 into the equation m=210g m = \frac{210}{g} m=g210​:m=21021=10 m = \frac{210}{21} = 10 m=21210​=10

STEP 24

d. Find g g g when m=15 m = 15 m=15.Substitute m=15 m = 15 m=15 into the equation m=210g m = \frac{210}{g} m=g210​:15=210g 15 = \frac{210}{g} 15=g210​Solve for g g g:g=21015=14 g = \frac{210}{15} = 14 g=15210​=14

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1. We are dealing with problems involving direct, inverse, and joint variation.
2. The problems require finding constants of variation, equations of variation, and solving for unknown variables.

1. Solve Problem 1: Direct Variation
2. Solve Problem 2: Inverse Variation
3. Solve Problem 3: Joint Variation
4. Solve Problem 4: Direct and Inverse Variation
5. Solve Problem 5: Inverse Variation

b. Write the equation of variation.
The equation is $y = 24x$ .

c. Find $y$ when $x = 12$ .
Substitute $x = 12$ into the equation $y = 24x$ :
$y = 24 \times 12 = 288$

d. Find $x$ when $y = 27$ .
Substitute $y = 27$ into the equation $y = 24x$ :
$27 = 24x$
Solve for $x$ :
$x = \frac{27}{24} = \frac{9}{8}$

b. Write the equation of variation.
The equation is $m = \frac{576}{n}$ .

c. Find $m$ when $n = 9$ .
Substitute $n = 9$ into the equation $m = \frac{576}{n}$ :
$m = \frac{576}{9} = 64$

d. Find $n$ when $m = 12$ .
Substitute $m = 12$ into the equation $m = \frac{576}{n}$ :
$12 = \frac{576}{n}$
Solve for $n$ :
$n = \frac{576}{12} = 48$

b. Write the equation of variation.
The equation is $a = 24bc$ .

c. Find $a$ when $b = 9$ and $c = 3$ .
Substitute $b = 9$ and $c = 3$ into the equation $a = 24bc$ :
$a = 24 \times 9 \times 3 = 648$

d. Find $b$ when $a = 720$ and $c = 6$ .
Substitute $a = 720$ and $c = 6$ into the equation $a = 24bc$ :
$720 = 24b \times 6$
Solve for $b$ :
$b = \frac{720}{144} = 5$

e. Find $c$ when $a = 960$ and $b = 8$ .
Substitute $a = 960$ and $b = 8$ into the equation $a = 24bc$ :
$960 = 24 \times 8 \times c$
Solve for $c$ :
$c = \frac{960}{192} = 5$

b. Write the equation of variation.
The equation is $a = 6 \frac{b}{c}$ .

c. Find $a$ when $b = 9$ and $c = 3$ .
Substitute $b = 9$ and $c = 3$ into the equation $a = 6 \frac{b}{c}$ :
$a = 6 \frac{9}{3} = 18$

d. Find $b$ when $a = 24$ and $c = 5$ .
Substitute $a = 24$ and $c = 5$ into the equation $a = 6 \frac{b}{c}$ :
$24 = 6 \frac{b}{5}$
Solve for $b$ :
$b = \frac{24 \times 5}{6} = 20$

e. Find $c$ when $a = 45$ and $b = 15$ .
Substitute $a = 45$ and $b = 15$ into the equation $a = 6 \frac{b}{c}$ :
$45 = 6 \frac{15}{c}$
Solve for $c$ :
$c = \frac{6 \times 15}{45} = 2$

b. Write the equation of variation.
The equation is $m = \frac{210}{g}$ .

c. Find $m$ when $g = 21$ .
Substitute $g = 21$ into the equation $m = \frac{210}{g}$ :
$m = \frac{210}{21} = 10$

d. Find $g$ when $m = 15$ .
Substitute $m = 15$ into the equation $m = \frac{210}{g}$ :
$15 = \frac{210}{g}$
Solve for $g$ :
$g = \frac{210}{15} = 14$