Help Examples on Equation and Formula

1. (1 pt) The distance (in miles) traveled when driving at a certain speed s for 24
hours, then driving 18 miles/hour faster for another 3 hours. Express the distance in
terms of s.

The distance formula is d = st where s is speed and t is time and d is distance.
For the first 24 hours, the speed is s, so the distance driven is 24s. For the next 3
hours, the speed is 18 miles/hour faster, so the new speed is s + 18. So the distance
driven in these three hours would be 3(s + 18) = 3s + 54. So the total driven distance
is 24s + 3s + 54 = 27s + 54.

4. One positive number is one-fifth of another number. The difference between the
two numbers is 108, find the numbers.

Let the first number be x, since it is one- fifth of the other number, the other number
must be 5x. The difference is 5x - x = 108. You can then solve this equation for x.
Enter your answer as x and 5x in that order since x is smaller.

14. The difference of two positive numbers is 5 and the sum of their squares is 97.
Find the numbers.

Let the smaller number be x so the bigger number would be x+5 since their difference
is 5. The sum of their squares is x^2 + (x + 5)^2 = 97 by the given condition. And
(x + 5)^2 = x^2 + 10x + 25. So the equation is simplified to 2x^2 + 10x + 25 = 97, or
2x^2+10x-72 = 0. Dividing both sides by 2 gives x^2+5x-36 = 0. You can now factor
this and solve the equation without having to use the quadratic formula.

15. NASA launches a rocket at t = 0 seconds. Its height, in meters above sea-level,
as a function of time is given by h(t) = -4.9t^2 + 133t + 171. Assuming that the rocket
will splash down into the ocean, at what time does splashdown occur?

When the rocket hits the ocean, its height would be zero since the height is measured
as over the sea-level. That is, h(t) = 0. So one obtains the equation 0 = -4.9t^2+133t+
171. This is a quadratic equation. Since the numbers are big, you definitely need to
use the quadratic formula. In this example, a = -4.9, b = 133 and c = 171. Use
your calculator. You get two answers but only the positive one makes sense as you
are looking for the time after launching when the rocket hits ocean which has to be a
positive number.