Question

2 - 7 practice
literal equations and dimensional analysis
solve each equation or formula for the variable indicated.

1. $\frac{d}{t}=r$, for $r$
2. $6w - y = 2z$, for $w$
3. $mx + 4y = 3t$, for $x$
4. $9s - 5g=-4u$, for $s$
5. $ab + 3c = 2x$, for $b$
6. $2p = kx - t$, for $x$
7. $\frac{2}{3}m + a = a + r$, for $m$
8. $\frac{2}{5}h + g = d$, for $h$
9. $\frac{2}{3}y + v = x$, for $y$
10. $\frac{3}{4}a - q = k$, for $a$
11. $\frac{rx + 9}{5}=h$, for $x$
12. $\frac{3b - 4}{2}=c$, for $b$
13. $2w - y = 7w - 2$, for $w$
14. $3\ell + y = 5 + 5\ell$, for $\ell$
15. electricity the formula for ohms law is $e = ir$, where $e$ represents voltage measured in volts, $i$ represents current measured in amperes, and $r$ represents resistance measured in ohms.

a. solve the formula for $r$.
b. suppose a current of 0.25 ampere flows through a resistor connected to a 12 - volt battery. what is the resistance in the circuit?

1. motion in uniform circular motion, the speed $v$ of a point on the edge of a spinning disk is $v = \frac{2\pi}{t}r$, where $r$ is the radius of the disk and $t$ is the time it takes the point to travel once around the circle.

a. solve the formula for $r$.
b. suppose a merry - go - round on a playground is spinning once every 3 seconds. if a point on the outside edge has a speed of 12.56 feet per second, what is the radius of the merry - go - round? (use 3.14 for $\pi$.)

1. highways interstate 90 is the longest interstate highway in the united states, connecting the cities of seattle, washington and boston, massachusetts. the interstate is 4,987,000 meters in length. if 1 mile = 1.609 kilometers, how many miles long is interstate 90?

Explanation:

Step1: Solve \(d = rt\) for \(r\)

Divide both sides of the equation \(d = rt\) by \(t\).
$$r=\frac{d}{t}$$

Step2: Solve \(6w - y=2z\) for \(w\)

Add \(y\) to both sides: \(6w=2z + y\). Then divide both sides by \(6\).
$$w=\frac{2z + y}{6}$$

Step3: Solve \(mx + 4y=3t\) for \(x\)

Subtract \(4y\) from both sides: \(mx=3t - 4y\). Then divide both sides by \(m\) (\(m
eq0\)).
$$x=\frac{3t - 4y}{m}$$

Step4: Solve \(9s - 5g=-4u\) for \(s\)

Add \(5g\) to both sides: \(9s=-4u + 5g\). Then divide both sides by \(9\).
$$s=\frac{-4u + 5g}{9}$$

Step5: Solve \(ab + 3c=2x\) for \(b\)

Subtract \(3c\) from both sides: \(ab=2x - 3c\). Then divide both sides by \(a\) (\(a
eq0\)).
$$b=\frac{2x - 3c}{a}$$

Step6: Solve \(2p=kx - t\) for \(x\)

Add \(t\) to both sides: \(kx=2p + t\). Then divide both sides by \(k\) (\(k
eq0\)).
$$x=\frac{2p + t}{k}$$

Step7: Solve \(\frac{2}{3}m + a=a + r\) for \(m\)

Subtract \(a\) from both sides: \(\frac{2}{3}m=r\). Then multiply both sides by \(\frac{3}{2}\).
$$m=\frac{3r}{2}$$

Step8: Solve \(\frac{2}{5}h+g = d\) for \(h\)

Subtract \(g\) from both sides: \(\frac{2}{5}h=d - g\). Then multiply both sides by \(\frac{5}{2}\).
$$h=\frac{5(d - g)}{2}$$

Step9: Solve \(\frac{2}{3}y + v=x\) for \(y\)

Subtract \(v\) from both sides: \(\frac{2}{3}y=x - v\). Then multiply both sides by \(\frac{3}{2}\).
$$y=\frac{3(x - v)}{2}$$

Step10: Solve \(\frac{3}{4}a - q=k\) for \(a\)

Add \(q\) to both sides: \(\frac{3}{4}a=k + q\). Then multiply both sides by \(\frac{4}{3}\).
$$a=\frac{4(k + q)}{3}$$

Step11: Solve \(\frac{rx + 9}{5}=h\) for \(x\)

Multiply both sides by \(5\): \(rx+9 = 5h\). Subtract \(9\) from both sides: \(rx=5h - 9\). Then divide both sides by \(r\) (\(r
eq0\)).
$$x=\frac{5h - 9}{r}$$

Step12: Solve \(\frac{3b - 4}{2}=c\) for \(b\)

Multiply both sides by \(2\): \(3b-4 = 2c\). Add \(4\) to both sides: \(3b=2c + 4\). Then divide both sides by \(3\).
$$b=\frac{2c + 4}{3}$$

Step13: Solve \(2w - y=7w - 2\) for \(w\)

Subtract \(2w\) from both sides: \(-y = 5w-2\). Add \(2\) to both sides: \(5w=2 - y\). Then divide both sides by \(5\).
$$w=\frac{2 - y}{5}$$

Step14: Solve \(3\ell+y = 5 + 5\ell\) for \(\ell\)

Subtract \(3\ell\) from both sides: \(y=5 + 2\ell\). Subtract \(5\) from both sides: \(2\ell=y - 5\). Then divide both sides by \(2\).
$$\ell=\frac{y - 5}{2}$$

Step15a: Solve \(E = IR\) for \(R\)

Divide both sides by \(I\) (\(I
eq0\)).
$$R=\frac{E}{I}$$

Step15b: Find \(R\) when \(E = 12\) volts and \(I=0.25\) ampere

Substitute \(E = 12\) and \(I = 0.25\) into \(R=\frac{E}{I}\).
$$R=\frac{12}{0.25}=48$$ ohms

Step16a: Solve \(v=\frac{2\pi}{t}r\) for \(r\)

Multiply both sides by \(\frac{t}{2\pi}\) (\(t
eq0\)).
$$r=\frac{vt}{2\pi}$$

Step16b: Find \(r\) when \(v = 12.56\) feet per second and \(t = 3\) seconds (\(\pi = 3.14\))

Substitute \(v = 12.56\), \(t = 3\), and \(\pi=3.14\) into \(r=\frac{vt}{2\pi}\).
$$r=\frac{12.56\times3}{2\times3.14}=6$$ feet

Step17: Convert \(4987000\) meters to miles

Since \(1\) mile \(=1.609\) kilometers \(=1609\) meters. Let \(x\) be the length in miles. Then \(x=\frac{4987000}{1609}\approx3099.44\) miles

Answer:

1. \(r=\frac{d}{t}\)
2. \(w=\frac{2z + y}{6}\)
3. \(x=\frac{3t - 4y}{m}\) (\(m

eq0\))

1. \(s=\frac{-4u + 5g}{9}\)
2. \(b=\frac{2x - 3c}{a}\) (\(a

eq0\))

1. \(x=\frac{2p + t}{k}\) (\(k

eq0\))

1. \(m=\frac{3r}{2}\)
2. \(h=\frac{5(d - g)}{2}\)
3. \(y=\frac{3(x - v)}{2}\)
4. \(a=\frac{4(k + q)}{3}\)
5. \(x=\frac{5h - 9}{r}\) (\(r

eq0\))

1. \(b=\frac{2c + 4}{3}\)
2. \(w=\frac{2 - y}{5}\)
3. \(\ell=\frac{y - 5}{2}\)

15a. \(R=\frac{E}{I}\) (\(I
eq0\))
15b. \(48\) ohms
16a. \(r=\frac{vt}{2\pi}\) (\(t
eq0\))
16b. \(6\) feet

1. Approximately \(3099.44\) miles