2 quadratics questions...?

Question

The height h (in feet) of an object that is dropped from the height of s feet is given by the formula h = s - 16t ^2 , where t is the time the object has been falling. A 6 foot tall woman on a sidewalk looks directly overhead and sees a window washer drop a bottle from the 5 story. How long does she have to get out of the way? Round to the nearest tenth. (A story is 12 feet.) Choose the answer from the following:

1.6 seconds

1.8 seconds

2.3 seconds

1.7 seconds


The hypotenuse of a right triangle is 2.8 units long. The longer leg is 1.8 units longer than the shorter leg. Find the lengths of the sides of the triangle.

2.8, 2.7, 0.9

2.6, 0.8, 2.8

0.7, 2.5, 2.8

2.8, 1.8, 4.7

Answer 1

PROBLEM 1:

The bottle starts at 60 ft (5 x 12 = 60).
s = 60

It has to fall until it is 6 ft. from the ground (the top of her head)
h = 6

6 = 60 - 16t²

Now solve for t:
-54 = -16t²
16t² = 54
t² = 54 / 16
t = sqrt(54/16)
t = sqrt(54) / 4
t = sqrt(9)*sqrt(6) / 4
t = 3 * sqrt(6) / 4
t ≈ 1.84 seconds

Answer B)
1.8 seconds

PROBLEM 2:

Let x be the length of the shorter leg
Let x + 1.8 be the length of the longer leg.

By the Pythagorean theoream:

x² + (x + 1.8)² = 2.8²

x² + (x² + 3.6x + 1.8²) = 2.8²
2x² + 3.6x + 1.8² - 2.8² = 0
2x² + 3.6x - 4.6 = 0

I'd multiply both sides by 5 to get rid of fractions:
10x² + 18x - 23 = 0

From the quadratic formula:
x = [ -18 ± sqrt( 18² - 4(10)(-23) ) ] / 20
x = [ -18 ± sqrt( 324 + 920 ) ] / 20
x = [ -18 ± sqrt( 1244 ) ] / 20
x = [ -18 ± 35.27 ] / 20

Obviously the length has to be positive so:
x = ( -18 + 35.27 ) / 20
x = 0.8635
x ≈ 0.9 (round up)

The longer leg is 0.9 + 1.8 = 2.7 approximately

Answer A)
Hypotenuse = 2.8
Longer leg ≈ 2.7
Shorter leg ≈ 0.9

Answer 2

let x be the shorter leg.
longer leg=x+1.8
2.8^2=x^2+(x+1.8)^2
7.84=x^2+x^2+3.6x+3.24
2x^2+3.6x-4.6=0
ax^2+bx+c=0
x=[-b+/-sqrt(b^2-4ac)]/2a
a=2 b=3.6 c=-4.6
b^2-4ac=(3.6)^2-4(2)(-4.6)
=12.96+36.8=49.76
x=[-3.6+sqrt(49.76)]/4
x=[-3.6-sqrt(49.76)]/4
.864
Now, the longer leg is 0.864+1.8=2.6 (approx)
2.6, .8, 2.8