MTH 112
Practice Problems for Test 3
Solve the problem.
1) If y varies directly as the square of x, and y = 480 when x = 8, find y when x = 2.
1)
2) If the resistance in an electrical circuit is held constant, the amount of current flowing
through the circuit is directly proportional to the amount of voltage applied to the circuit.
When 6 volts are applied to a circuit, 60 milliamperes of current flow through the circuit.
Find the new current if the voltage is increased to 14 volts.
2)
3) x varies inversely as y 2 , and x = 5 when y = 24. Find x when y = 6.
3)
4) If the voltage, V, in an electric circuit is held constant, the current, I, is inversely
proportional to the resistance, R. If the current is 420 milliamperes when the resistance is 2
ohms, find the current when the resistance is 12 ohms.
4)
Solve.
Write an equation that expresses the relationship. Use k for the constant of proportionality.
5) s varies jointly as t and u and inversely as the square root of a.
Write an equation that expresses the relationship. Use k as the constant of variation.
6) The weight of a body above the surface of the earth is inversely proportional to the square
of its distance from the center of the earth. What is the effect on the weight when the
distance is multiplied by 3?
Find the variation equation for the variation statement.
7) t varies directly as r and inversely as s; t = 2 when r = 18 and s = 72
5)
6)
7)
Solve the problem.
8) f varies jointly as q2 and h, and f = 54 when q = 3 and h = 2. Find f when q = 2 and h = 3.
8)
9) The amount of paint needed to cover the walls of a room varies jointly as the perimeter of
the room and the height of the wall. If a room with a perimeter of 80 feet and 6-foot walls
requires 4.8 quarts of paint, find the amount of paint needed to cover the walls of a room
with a perimeter of 30 feet and 10-foot walls.
9)
Approximate the number using a calculator. Round your answer to three decimal places.
10) 6 1.8
Solve the problem.
11) A city is growing at the rate of 0.7% annually. If there were 2,899,000 residents in the city
in 1992, find how many (to the nearest ten-thousand) are living in that city in 2000. Use
y = 2,899,000(2.7)0.007t.
1
10)
11)
Graph the function by making a table of coordinates.
5 x
12) f(x) = 2
6
12)
y
4
2
-6
-4
-2
2
4
6 x
-2
-4
-6
The graph of an exponential function is given. Select the function for the graph from the functions listed.
13)
y
10
5
-10
-5
5
10
x
-5
-10
A) f(x) = 2 x - 2
B) f(x) = 2 x - 2
C) f(x) = 2 x
D) f(x) = 2 x + 2
Approximate the number using a calculator. Round your answer to three decimal places.
14) e2.8
Solve the problem.
15) The size of the bear population at a national park increases at the rate of 4.1% per year. If
the size of the current population is 173, find how many bears there should be in 4 years.
Use the function f(x) = 173e0.041t and round to the nearest whole number.
Use the compound interest formulas A = P 1 + 14)
15)
r nt
and A = Pe rt to solve.
n
16) Find the accumulated value of an investment of $700 at 16% compounded quarterly for 2
years.
16)
17) Find the accumulated value of an investment of $5000 at 8% compounded continuously
for 4 years.
17)
2
13)
Write the equation in its equivalent exponential form.
18) log 64 = 3
4
18)
19) log 49 = 2
b
19)
Write the equation in its equivalent logarithmic form.
20) 2 3 = x
20)
21) 4 x = 16
21)
Evaluate the expression without using a calculator.
22) log 8
2
22)
1
2 4
23)
24) log 1
6
24)
23) log
25) 6
log 14
6
25)
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
26) log (7 · 3)
26)
5
27) log (27x)
3
27)
28) log
7
6 5
28)
29) log x
10
29)
30) logg x4
30)
31) ln 3
x
31)
32) logb (yz 8 )
33) log
32)
x5
4 y8
33)
3
34) log
x
8
2
34)
35) log m + log n
c
c
35)
36) log (x - 3) - log (x - 5)
4
4
36)
37) ln x + 5ln y
37)
38) 3 log x + 5 log (x - 6)
6
6
38)
Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places
39) log 9
6
40) log
13
74.7
39)
40)
Solve the equation by expressing each side as a power of the same base and then equating exponents.
1
41) 4 (7 - 3x) = 41)
16
42) 25x + 1 = 125 x - 3
42)
Solve the exponential equation. Express the solution set in terms of natural logarithms.
43) 5 8x = 4.5
44) e5x = 3
43)
44)
Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the
solution.
45) 3 x + 6 = 7
45)
46) e3x - 10 - 9 = 1334
46)
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
47) log x = 4
47)
3
48) ln x = 6
48)
49) log 7 + log x = 1
9
9
49)
50) log 3 (x + 6) + log 3 (x - 6) - log 3 x = 2
50)
4
51) log (x + 5) = 2 + log (x - 4)
5
5
51)
52) log 4x = log 5 + log (x - 1)
52)
53) The value of a particular investment follows a pattern of exponential growth. In the year
2000, you invested money in a money market account. The value of your investment t
years after 2000 is given by the exponential growth model A = 4100e0.056t. How much did
53)
Solve.
you initially invest in the account?
54) The population of a particular country was 21 million in 1983; in 1997, it was 27 million.
The exponential growth function A =21ekt describes the population of this country t years
54)
after 1983. Use the fact that 14 years after 1983 the population increased by 6 million to
find k to three decimal places.
REVIEW MATERIAL
Find and simplify the difference quotient f(x + h) - f(x)
, h≠ 0 for the given function.
h
55) f(x) = 8x2
55)
Evaluate the piecewise function at the given value of the independent variable.
56)
x2 - 6
if x ≠ -4
g(x) = x + 4
56)
x + 2 if x = -4
Determine g(-2).
Answer the question.
57) How can the graph of f(x) = 1
1
- 2 be obtained from the graph of y = ?
x + 6
x
57)
A) Shift it horizontally 6 units to the right. Stretch it vertically by a factor of 2.
1
B) Shrink it horizontally by a factor of . Shift it 2 units down.
2
C) Shift it horizontally 6 units to the left. Shift it 2 units down.
D) Shift it horizontally 6 units to the left. Shift it 2 units up.
Complete the square and write the equation in standard form. Then give the center and radius of the circle.
58) x2 + y 2 + 14x - 2y = -34
58)
Divide using synthetic division.
-2x3 - 10x2 + 9x - 18
59)
x + 6
59)
5
Use the vertex and intercepts to sketch the graph of the quadratic function.
60) f(x) = x2 + 2x - 8
60)
y
10
5
-10
-5
5
10
x
-5
-10
Find a rational zero of the polynomial function and use it to find all the zeros of the function.
61) f(x) = 3x3 - 17x2 + 18x + 8
Use the Leading Coefficient Test to determine the end behavior of the polynomial function.
62) f(x) = 2x4 - 5x3 + 5x2 - 4x + 5
A) rises to the left and rises to the right
C) falls to the left and rises to the right
61)
62)
B) falls to the left and falls to the right
D) rises to the left and falls to the right
Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval
notation.
63) x2 - 2x - 8 ≤ 0
63)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval
notation.
x - 3
64)
< 0
64)
x + 8
6
Answer Key
Testname: TEST 3 PRACTICE PROBLEMS FA10
1)
2)
3)
4)
30
140 milliamperes
x = 80
70 milliamperes
ktu
5) s = a
6) The weight is divided by 9
8r
7) t = s
8) f = 36
9) 3 quarts
10) 25.158
11) 3,060,000
12)
6
y
4
2
-6
-4
-2
2
4
6 x
-2
-4
-6
13) C
14) 16.445
15) 204
16) $958.00
17) $6885.64
18) 4 3 = 64
19) b2 = 49
20) log x = 3
2
21) log 16 = x
4
22) 3
23) -2
24) 0
25) 14
26) log 7 + log 3
5
5
27) 3 + log x
3
28) log 7 - log 5
6
6
29) log x - 1
30) 4logg x
31)
1
ln x
3
32) logb y + 8 logb z
7
Answer Key
Testname: TEST 3 PRACTICE PROBLEMS FA10
33) 5 log x - 8 log y
4
4
1
34) log x - 3
2
2
35) log (mn)
c
x - 3
36) log
4 x - 5
37) ln xy5
38) log x3 (x - 6)5
6
39) 1.2263
40) 1.6817
41) {3}
42) 11
ln 4.5
43)
8 ln 5
44)
ln 3
5
45) -4.23
46) 5.73
47) {81}
48) e6
9
49) { }
7
50) {12}
35
51)
8
52) {5}
53) $4100.00
54) 0.018
55) 8(2x+h)
56) - 1
57) C
58) (x + 7)2 + (x - 1)2 = 16
(-7, 1), r = 4
59) -2x2 + 2x - 3
60)
y
10
5
-10
-5
5
10
x
-5
-10
8
Answer Key
Testname: TEST 3 PRACTICE PROBLEMS FA10
1
61) - , 2, 4
3
62) A
63) [-2, 4]
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
64) (-8, 3)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
9