The distance formula is essential in this problem.

$\displaystyle Distance=rate\times time$

First, use Kate's info to figure out the rate for both girls since they're travelling at the same speed, which is $\displaystyle 60$. Then, plug in that rate to the formula with Bella's information, which gives her a time of $\displaystyle 7.5$ hours. 

At 45 mph, Joe drove 60 miles in $\displaystyle 60 \div 45 = \frac{60}{45}= \frac{4}{3}$ hours.

At 60 mph, he drove 30 miles in $\displaystyle 30 \div 60 = \frac{1}{2}$ hours. 

He made the 90-mile trip in $\displaystyle \frac{4}{3} + \frac{1}{2} = \frac{8}{6} + \frac{3}{6} = \frac{11}{6}$ hours, so divide 90 by $\displaystyle \frac{11}{6}$ to get the average speed in mph:

$\displaystyle 90 \div \frac{11}{6} = \frac{90}{1} \div \frac{11}{6} = \frac{90}{1} \cdot \frac{6}{11} = \frac{540} {11} \approx49$

Using the conversion formula, you would multiply 5.3 miles by 5280 feet and you will get 27,984 feet.

Using d = rt, we know that first part of the trip can be represented by f = rg.  The second part of the trip can be represented by 30 = rx, where x is some unknown number of hours.  Note that the rate r is in both equations because Sophie is traveling at the same rate as mentioned in the problem.

Solve each equation for the time (g in equation 1, x in equation 2).

g = f/r

x = 30/r

The total time is the sum of these two times

Note that, from equation 1, r = f/g, so 

=

Traveling 20 mph faster than Gary, it will take the officers 30 minutes to catch up to Gary. The answer is 3:30 PM.

To find this answer just do total miles divided by miles per gallon in order to find how many gallons of gas it will take to get from point A to Point B. Then multiply that answer by the cost of gasoline per gallon to find total amount spent on gasoline.

In the first three hours, he travels 180 miles.

$\displaystyle 3\times60=180$

In the next two hours, he travels 144 miles.

$\displaystyle 2\times 72=144$

for a total of 324 miles.

$\displaystyle 180+144=324$

Divide by the total number of hours to obtain the average traveling speed.

$\displaystyle \frac{324}{5}=64.8\ mph$

Tom runs a 100m race in a certain amount of time.  If John runs the same race, he takes 2 seconds longer.  If John ran at 8m/s, how fast did Tom run?

Let $\displaystyle x$ denote the amount of time that it took Tom to run the race.  Then it took John $\displaystyle x+2$ seconds to run the same race going 8m/s.  At 8m/s, it takes 12.5 seconds to finish a 100m race.  This means it took Tom 10.5 seconds to finish.  Running 100m in 10.5 seconds is the same as $\displaystyle 100/10.5 \approx 9.5m/s$

Write the distance formula.

$\displaystyle d= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Substitute the values of the points into the formula.

$\displaystyle d= \sqrt{(6-2)^2+(7-3)^2}$

$\displaystyle d= \sqrt{(4)^2+(4)^2} = \sqrt{16+16} = \sqrt{32}$

The square root of $\displaystyle 32$ can be reduced because $\displaystyle 16$, a factor of $\displaystyle 32$, is a perfect square. $\displaystyle 16*2=32$. 

Now we have $\displaystyle d=\sqrt{16*2}=4\sqrt{2}$

Write the distance formula.

$\displaystyle D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Plug in the points.

$\displaystyle D=\sqrt{(2-0)^2+(0-2)^2} = \sqrt{(2)^2+(-2)^2} = \sqrt{4+4} = \sqrt{8}$

$\displaystyle D= 2\sqrt2$

The distance is: $\displaystyle 2\sqrt2$