Quiz 1
Answer keys
Fall 2013
Math 0400
1. [5 points] Write an equation of the circle that passes through the point (5, 2) and has center
at (2, −3).
Solution: The radius of the circle is the distance between its center and the point (5, 2). [1 pt]
Then r2 = (5 − 2)2 + (2 + 3)2 = 9 + 25 = 34.
[2 pts]
The equation of the circle is (x − 2)2 + (y + 3)2 = r2 or (x − 2)2 + (y + 3)2 = 34.
Answer: (x − 2)2 + (y + 3)2 = 34.
[2 pts]
2. [5 points] If the line passing through the points (1, a) and (5, 8) is parallel to the line passing
through the points (4, 9) and (1, a + 2) what is the value of a?
Solution:
Lines are parallel and therefore their slopes are equal:
a+2−9
8−a
=
5−1
1−4
⇔
[1 pt]
8−a
a−7
=
⇔ −3(8 − a) = 4(a − 7) ⇔ −24 + 3a = 4a − 28
4
−3
[2 pts]
⇔ −24 + 28 = 4a − 3a ⇔ a = 4
Answer: a = 4.
[2 pts]
3. [5 points] A manufacturer has a monthly fixed cost of $100, 000 and a production cost of
$14 for each unit produced. The product sells for $20/unit
1. What is the cost function?
2. What is the revenue function?
3. What is the profit function?
4. Compute the profit (loss) corresponding to production level of 15, 000 units.
Solution:
The cost function is C(x) = 14x + 100000,
the revenue function is R(x) = 20x,
[1 pt]
[1 pt]
the profit function is P (x) = R(x) − C(x) = 20x − 14x − 100000 = 6x − 100000.
1
[1 pt]
P (15000) = 6 · 15000 − 100000 = 90000 − 100000 = −10000.
Hence to production level of 15, 000 units corresponds to a loss of $10, 000.
[2 pts]
bonus problem [5 points extra] Find an equation of the line that passes through the point
(1, −2) and is perpendicular to the line passing through the points (−2, −1) and (4, 3).
Solution:
The slope of the second line is
3
Hence the slope of the first line is − .
2
3+1
4
2
= = .
4+2
6
3
[1 pt]
[2 pts]
3
In the point-slope form its equation is y + 2 = − (x − 1)
2
3
3
3
1
In the slope-intercept form the equation is y = − x + − 2 ⇔ y = − x − .
2
2
2
2
Answer:
3
either y + 2 = − (x − 1)
2
1
3
or
y =− x−
2
2
[2 pts]
2