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Mathematics LibreTexts

3.1E: Exercises

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    Practice Makes Perfect

    Use the Distance, Rate, and Time Formula

    In the following exercises, solve.

    Exercise \(\PageIndex{1}\)

    Steve drove for 8\(\frac{1}{2}\) hours at 72 miles per hour. How much distance did he travel?

    Exercise \(\PageIndex{2}\)

    Socorro drove for 4\(\frac{5}{6}\) hours at 60 miles per hour. How much distance did she travel?

    Answer

    290 miles

    Exercise \(\PageIndex{3}\)

    Yuki walked for 1\(\frac{3}{4}\) hours at 4 miles per hour. How far did she walk?

    Exercise \(\PageIndex{4}\)

    Francie rode her bike for 2\(\frac{1}{2}\) hours at 12 miles per hour. How far did she ride?

    Answer

    30 miles

    Exercise \(\PageIndex{5}\)

    Connor wants to drive from Tucson to the Grand Canyon, a distance of 338 miles. If he drives at a steady rate of 52 miles per hour, how many hours will the trip take?

    Exercise \(\PageIndex{6}\)

    Megan is taking the bus from New York City to Montreal. The distance is 380 miles and the bus travels at a steady rate of 76 miles per hour. How long will the bus ride be?

    Answer

    5 hours

    Exercise \(\PageIndex{7}\)

    Aurelia is driving from Miami to Orlando at a rate of 65 miles per hour. The distance is 235 miles. To the nearest tenth of an hour, how long will the trip take?

    Exercise \(\PageIndex{8}\)

    Kareem wants to ride his bike from St. Louis to Champaign, Illinois. The distance is 180 miles. If he rides at a steady rate of 16 miles per hour, how many hours will the trip take?

    Answer

    11.25 hours

    Exercise \(\PageIndex{9}\)

    Javier is driving to Bangor, 240 miles away. If he needs to be in Bangor in 4 hours, at what rate does he need to drive?

    Exercise \(\PageIndex{10}\)

    Alejandra is driving to Cincinnati, 450 miles away. If she wants to be there in 6 hours, at what rate does she need to drive?

    Answer

    75 mph

    Exercise \(\PageIndex{11}\)

    Aisha took the train from Spokane to Seattle. The distance is 280 miles and the trip took 3.5 hours. What was the speed of the train?

    Exercise \(\PageIndex{12}\)

    Philip got a ride with a friend from Denver to Las Vegas, a distance of 750 miles. If the trip took 10 hours, how fast was the friend driving?

    Answer

    75 mph

    Solve a Formula for a Specific Variable

    In the following exercises, use the formula \(d=rt\).

    Exercise \(\PageIndex{13}\)

    Solve for \(t\)

    1. when \(d=350\) and \(r=70\)
    2. in general

    Exercise \(\PageIndex{14}\)

    Solve for \(t\)

    1. when \(d=240\) and \(r=60\)
    2. in general
    Answer
    1. \(t=4\)
    2. \(t=\frac{d}{r}\)

    Exercise \(\PageIndex{15}\)

    Solve for \(t\)

    1. when \(d=510\) and \(r=60\)
    2. in general

    Exercise \(\PageIndex{16}\)

    Solve for \(t\)

    1. when \(d=175\) and \(r=50\)
    2. in general
    Answer
    1. \(t=3.5\)
    2. \(t=\frac{d}{r}\)

    Exercise \(\PageIndex{17}\)

    Solve for \(r\)

    1. when \(d=204\) and \(t=3\)
    2. in general

    Exercise \(\PageIndex{18}\)

    Solve for \(r\)

    1. when \(d=420\) and \(t=6\)
    2. in general
    Answer
    1. \(r=70\)
    2. \(r=\frac{d}{t}\)

    Exercise \(\PageIndex{19}\)

    Solve for \(r\)

    1. when \(d=160\) and \(t=2.5\)
    2. in general

    Exercise \(\PageIndex{20}\)

    Solve for \(r\)

    1. when \(d=180\) and \(t=4.5\)
    2. in general
    Answer
    1. \(r=40\)
    2. \(r=\frac{d}{t}\)

    In the following exercises, use the formula \(A=\frac{1}{2} b h\)

    Exercise \(\PageIndex{21}\)

    Solve for \(b\)

    1. when \(A=126\) and \(h=18\)
    2. in general

    Exercise \(\PageIndex{22}\)

    Solve for \(h\)

    1. when \(A=176\) and \(b=22\)
    2. in general
    Answer
    1. \(h=16\)
    2. \(h=\frac{2 A}{b}\)

    Exercise \(\PageIndex{23}\)

    Solve for \(h\)

    1. when \(A=375\) and \(b=25\)
    2. in general

    Exercise \(\PageIndex{24}\)

    Solve for \(b\)

    1. when \(A=65\) and \(h=13\)
    2. in general
    Answer
    1. \(b=10\)
    2. \(b=\frac{2 A}{h}\)

    In the following exercises, use the formula \(I = Prt\).

    Exercise \(\PageIndex{25}\)

    Solve for the principal, \(P\) for

    1. \(I=$5,480\), \(r=4\%\), \(t=7\) years
    2. in general

    Exercise \(\PageIndex{26}\)

    Solve for the principal, \(P\) for

    1. \(I=$3,950\), \(r=6\%\), \(t=5\) years
    2. in general
    Answer
    1. \(P=\$ 13,166.67\)
    2. \( P=\frac{I}{r t}\)

    Exercise \(\PageIndex{27}\)

    Solve for the time, \(t\) for

    1. \(I=$2,376\), \(P=$9,000\), \(r=4.4\%\)
    2. in general

    Exercise \(\PageIndex{28}\)

    Solve for the time, \(t\) for

    1. \(I=$624\), \(P=$6,000\), \(r=5.2\%\)
    2. in general
    Answer
    1. \(t=2\) years
    2. \(t=\frac{I}{Pr}\)

    In the following exercises, solve.

    Exercise \(\PageIndex{29}\)

    Solve the formula \(2x+3y=12\) for \(y\)

    1. when \(x=3\)
    2. in general

    Exercise \(\PageIndex{30}\)

    Solve the formula \(5x+2y=10\) for \(y\)

    1. when \(x=4\)
    2. in general
    Answer
    1. \(y=−5\)
    2. \(y=\frac{10-5 x}{2}\)

    Exercise \(\PageIndex{31}\)

    Solve the formula \(3x−y=7\) for \(y\)

    1. when \(x=−2\)
    2. in general

    Exercise \(\PageIndex{32}\)

    Solve the formula \(4x+y=5\) for \(y\)

    1. when \(x=−3\)
    2. in general
    Answer
    1. \(y=17\)
    2. \(y=5−4x\)

    Exercise \(\PageIndex{33}\)

    Solve \(a+b=90\) for \(b\).

    Exercise \(\PageIndex{34}\)

    Solve \(a+b=90\) for \(a\)

    Answer

    \(a=90-b\)

    Exercise \(\PageIndex{35}\)

    Solve \(180=a+b+c\) for \(a\)

    Exercise \(\PageIndex{36}\)

    Solve \(180=a+b+c\) for \(c\)

    Answer

    \(c=180-a-b\)

    Exercise \(\PageIndex{37}\)

    Solve the formula \(8 x+y=15\) for \(y\)

    Exercise \(\PageIndex{38}\)

    Solve the formula \(9 x+y=13\) for \(y\)

    Answer

    \(y=13-9 x\)

    Exercise \(\PageIndex{39}\)

    Solve the formula \(-4 x+y=-6\) for \(y\)

    Exercise \(\PageIndex{40}\)

    Solve the formula \(-5 x+y=-1\) for \(y\)

    Answer

    \(y=-1+5 x\)

    Exercise \(\PageIndex{41}\)

    Solve the formula \(4 x+3 y=7\) for \(y\)

    Exercise \(\PageIndex{42}\)

    Solve the formula \(3 x+2 y=11\) for \(y\)

    Answer

    \(y=\frac{11-3 x}{2}\)

    Exercise \(\PageIndex{43}\)

    Solve the formula \(x-y=-4\) for \(y\)

    Exercise \(\PageIndex{44}\)

    Solve the formula \(x-y=-3\) for \(y\)

    Answer

    \(y=3+x\)

    Exercise \(\PageIndex{45}\)

    Solve the formula \(P=2 L+2 W\) for \(L\)

    Exercise \(\PageIndex{46}\)

    Solve the formula \(P=2 L+2 W\) for \(W\)

    Answer

    \(W=\frac{P-2 L}{2}\)

    Exercise \(\PageIndex{47}\)

    Solve the formula \(C=\pi d\) for \(d\)

    Exercise \(\PageIndex{48}\)

    Solve the formula \(C=\pi d\) for \(\pi\)

    Answer

    \(\pi=\frac{C}{d}\)

    Exercise \(\PageIndex{49}\)

    Solve the formula \(V=L W H\) for \(L\)

    Exercise \(\PageIndex{50}\)

    Solve the formula \(V=L W H\) for \(H\)

    Answer

    \(H=\frac{V}{L W}\)

    Everyday Math

    Exercise \(\PageIndex{51}\)

    Converting temperature While on a tour in Greece, Tatyana saw that the temperature was 40o Celsius. Solve for F in the formula \(C=\frac{5}{9}(F−32)\) to find the Fahrenheit temperature.

    Exercise \(\PageIndex{52}\)

    Converting temperature Yon was visiting the United States and he saw that the temperature in Seattle one day was 50oFahrenheit. Solve for C in the formula \(F=\frac{9}{5}C+32\) to find the Celsius temperature.

    Answer

    \(10^{\circ} \mathrm{C}\)

    Writing Exercises

    Exercise \(\PageIndex{53}\)

    Solve the equation \(2x+3y=6\) for \(y\)

    1. when \(x=−3\)
    2. in general
    3. Which solution is easier for you, 1 or 2? Why?

    Exercise \(\PageIndex{54}\)

    Solve the equation \(5x−2y=10\) for \(x\)

    1. when \(y=10\)
    2. in general
    3. Which solution is easier for you, 1 or 2? Why?
    Answer

    Answers will vary.

    Self Check

    ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This is a table that has three rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads “use the distance, rate, and time formula,” and “solve a formula for a specific variable.” The rest of the cells are blank.

    ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?