Word Problems

1. What is the height (above ground level) when the object smacks into the ground? Well, zero, obviously. So I'm looking for the time when the height is s = 0. I'll set s equal to zero, and solve:

0 = –4.9t2 + 19.6t + 58.8
0 = t2 – 4t – 12
0 = (t – 6)(t + 2)

Then t = 6 or t = –2.

The second solution is from two seconds before launch, which doesn't make sense in this context. (It makes sense on the graph, because the line crosses the x-axis at –2, but negative time won't work in this word problem.) So "t = –2" is an extraneous solution, and I'll ignore it.

The object strikes the ground six seconds after launch.

2. An object is launched from ground level directly upward at 39.2 m/s. For how long is the object at or above a height of 34.3 meters?

The units this time are "meters", so the gravity number will be "4.9". Since the object started at ground level, the initial height was 0. Then my equation is:

Since this is a negative quadratic, the graph is an upside-down parabola. I can find the two times when the object is exactly 34.3 meters high, and I know that the object will be above 34.3 meters the whole time in between. Why "two time", and how do I know that the time period is between those two times? Because the first time will be when the object passes a height of 34.3 meters on its way up to its maximum height, and the second time when be when it passes 34.3 meters as it is falling back down to the ground. So I have to solve the following:

Then the object is at 34.3 meters at one second after launch (going up) and againt at seven seconds after launch (coming back down). Subtracting to find the difference, I find that:

s(t) = –4.9t2 + 39.2t
–4.9t2 + 39.2t = 34.3
t2 – 8t + 7 = 0
(t – 7)(t – 1) = 0
The object is at or above 34.3 meters for six seconds.

3. The three sides of a right-angled triangle are x, x+1 and 5. Find x and the area, if the longest side is 5.
The hypotenuse = 5
x2 + (x+1)2 = 52 (Pythagoras' Theorem)
x2 + x2 + 2x + 1 = 25
-25 => x2 + x2 + 2x - 24 = 0
(x + 6)(x - 4) = 0
(x + 6) = 0 or (x - 4) = 0
x = -6 or x = 4
x = 4;
Area = 0.5 x 3 x 4 = 6cm2

4. The product of two consecutive negative integers is 1122. What are the numbers?

Remember that consecutive integers are one unit apart, so my numbers are n and n + 1. Multiplying to get the product, I get:

The solutions are n = –34 and n = 33. I need a negative value, so I'll ignore "n = 33" and
take n = –34. Then the other number is n + 1 = (–34) + 1 = –33.

The two numbers are –33 and –34.