1.) The national cost "C" in billions of dollars for dental services can be modeled by the linear equation:
C = 2.85n+ 30.52
where n is the number of years since 1990 (Health Care Financing Administration, www.hcfa.gov).
a) Find and interpret the C-intercept for the line.
Put n = 0, we will get
C = 30.52
C-intercept is (0, 30.52)
b) Find and interpret the n-intercept for the line.
Put C = 0, we will get
0 = 2.85n + 30.52
2.85n = - 30.52
n = -30.52/2.85 = -10.71(approximately)
n-intercept is (-10.71, 0)
c) Graph the line for n ranging from 0 through 20.
2.) Demand equation. Helen's Health Foods usually sells 400 cans of ProPac Muscle Punch per week when the price is $5 per can. After experimenting with prices for some time, Helen has determined that the weekly demand can be found by using the equation:
d=600-40p
where d is the number of cans and p is the price per can.
a) Will Helen sell more or less Muscle Punch if she raises
her price from $5?
If she raises the price from $5, the demand will be less.
Let us take p = $6, then
d = 600 - 40*6 = 600 - 240 = 360
Thus the sales will decrease.
b) What happens to her sales every time she raises her
price by $1?
At every $1 increase in price the sales will decrease by 40.
c) Graph the equation.
3.) Marginal revenue. A defense attorney charges her client $4000 plus $120 per hour. The formula R=120n+4000 gives her revenue in dollars for n hours of work.
a) What is her revenue for 100 hours of work?
Put n = 100, we will get
R = 120*100 + 4000 = 12000 + 4000 = 16000
b) What is her revenue for 101 hours of work?
Put n = 101, we will get
R = 120*101 + 4000 = 12120 + 4000 = 16120
c) By how much did the one extra hour of work increase the revenue? (The increase in revenue is called the marginal revenue for the 101st hour.)
Increase = 16120 - 16000 = 120
4.) Gas laws. The volume of a gas is inversely proportional to the pressure on the gas. If the volume is 6 cubic centimeters when the pressure on the gas is 8 kilograms per square centimeter, then what is the volume when the pressure is 12 kilograms per square centimeter?
Let the volume be V and pressure be d then V = kd
Given, if the volume is 6 cubic centimeters when the pressure on the gas is 8 kilograms per square centimeter.
6 = 8k
=> k = 6/8 = ¾
Now, we have to find the volume when the pressure is 12 kilograms per square centimeter.
d = 12, k = ¾
V = kd = (3/4)*12 = 9 cubic centimeters.
5.) Graph the linear inequality:
2x<3y+6
6.) Solve each system by substitution. Determine whether the equations are independent, dependent or inconsistent.
a) y=-3x+19
y=2x-1
Substitute value of y = 2x - 1 in y= -3x+19, we will get
2x - 1 = -3x + 19
5x = 20
X = 4
Now substitute value of x in y = 2x - 1
y = 2*4 - 1 = 8 - 1 = 7
Solution: (4, 7)
System is consistent and independent as it has unique solution.
b) y= -4x -7
y=3x
Substitute y = 3x in y = -4x - 7, we will get
3x = -4x - 7
7x = -7
x = -1
y = 3x = -3
The solution is (-1, -3).
System is consistent and independent as it has unique solution.
7) Solve each system by Addition Method
a) 3x+5y = -11-----------(i)
x-2y = 11-------------(ii)
Mutiply equation (ii) by -3
-3x + 6y = -33
And add to equation (i)
3x + 5y = -11
11y = -44
y = -4
Given x - 2y = 11
x + 8 = 11
x = 3
The solution is (3, -4).
b) 2x=2-4
3x+y=-1
EQUATION NOT CLEAR
8) Solve each system by the Addition Method. Determine whether the equations are independent, dependent or inconsistent.
a) x-y=3
-6x+6y=17
6x - 6y = 18
-6x + 6y = 17
No solution.
Inconsistent and independent
9) Solve each system by Addition Method
a)
Multiply both equations by 63, we will get
27x + 35y = 1701---------(iii)
7x + 18y = 441----------(iv)
Multiply now (iii) by 7 and (iv) by -27
189x + 245y = 11907
-189x - 486y = - 11907
After adding we will get y = 0
(3/7)x + 0 = 27
x = 63
Solution (63, 0)
b) 3x-2.5y=7.125 --------------(i)
2.5x-3y=7.3125-----------------(ii)
Multiply equation (i) by -2.5
-7.5x + 6.25 y = - 17.8125
Multiply equation (ii) by 3
7.5x - 9y=7.3125 = 21.9375
add
-2.75y = 4.125
y = -1.5
2.5x + 4.5 = 7.3125
2.5x = 2.8125
x = 1.125
Solution (1.125, -1.5)