Detailed solutions to # 26 and 27 on study guide for exam 1
26. Solve: x
2
3
 4x
1
3
 12  0
This is an equation of the quadratic type: Set u = x
1
3
The rewrite the equation as u 2  4u  12  0
Factor to get (u + 6) (u – 2) = 0. Solve for u: u = -6 or u = 2
Plug in x
1
3
for u :
1
x 3 = -6 or x
1
Cube both sides of x 3 = -6
1
(x 3 ) 3 = (-6 )3
or
1
3
and x
=2
1
3
=2
1
(x 3 ) 3 = 23
Then x = - 216 or x = 8 Put answers in a set: {-216, 8}
27. A bullet is fired into the air with an initial upward velocity of 112 feet per second from the top of a
building 128 feet high. The equation that gives the height of the bullet at any time t is.
h  128  112t  16t 2 (This is correct, original equation was not!)
Find the times at which the bullet will be 288 feet in the air.
Since h gives the height of the bullet at any time t, we need to replace h by 288 and solve the resulting
equation for t.
288  128  112t  16t 2
Set the equation equal to 0:
Divide both sides by 16: t 2  7t  10  0
Factor: (t – 5) (t – 2) = 0 , so t = 5 or t = 2
Answer: 2 seconds and 5 seconds.
See next page for 2 examples.
16t 2  112t  160  0
Example: ( Like # 26) Solve
.
The substitution
puts the equation in a form
that is more obviously quadratic,
, which we can factor.
Then
or
But recall that
, so
corresponds to
,
corresponds to
.
To eliminate the fifth-roots in the two equations,
raise the equations to the fifth powers.
or
or
The solution set is { -1, 32}
.
Example (like # 27) A projectile is launched into the air with an initial upward velocity of 48 feet per
second from a building 32 feet high. The height (h) in feet of the projectile can be modeled by
h = - 16t2+ 48t + 32, where t is the time in seconds after it was launched. How much time, in seconds,
does it take the object to reach 64 feet above the ground?
Plug in 64 for h in h = - 16t2+ 48t + 32
then 64 = - 16t2+ 48t + 32
Set the quadratic equation equal to 0. 16t2 - 48t + 32 = 0
Divide the equation by 16.
t 2- 3t + 2 = 0. Factor: (t – 2) (t – 1) = 0 so t = 1 or t= 2
Solution: After 1 second and after 2 seconds.