The period of a pendulum, T, is the time it takes for the pendulum to make one back-and-forth swing, and is given by T = 2π
√
_ 9.8 L , where L is the length of the pendulum in meters. Find the length of a pendulum whose period is 3 seconds.SOLUTION Substitute 3 for T in the formula and solve for L. 3 = 2π
√
_
L 9.8 3_
2π =√
L_
9.8 Isolate the variable.
(
_
3 2π)
2 =(
√
_
L 9.8)
2Square both sides.
_
94π 2 =
_
9.8L Simplify.9.8
(
_
94π 2
)
= L Isolate L.2.23 ≈ L
The length of the pendulum is about 2.23 meters.
Lesson Practice
Solve each equation.
a. √ 2x + 3 = 21 b. 11 = √ x - 7 - 2 c. √ 3 2x + 1 = 0 d. 4 √ 3 3x - 4 = 16 e. √ 9x = √ 5x + 2 f. - √ 3 2x = √ 34x + 7 g. √ 3x + √ 3 x - 4 - 5 = 0
h. √ -1 - 4x = √ x - 11 i. x = √ 6 - 5x
j. The period of a pendulum, T, is the time it takes for the pendulum to make one back-and-forth swing, and is given by T = 2π
√
_ 9.8 L , where L is the length of the pendulum in meters. Find the length of a pendulum whose period is 5 seconds.(Ex 1)
(Ex 1) (Ex 1)(Ex 1)
(Ex 2)
(Ex 2) (Ex 2)(Ex 2)
(Ex 3)
(Ex 3) (Ex 3)(Ex 3)
(Ex 3) (Ex 3) (Ex 4)
(Ex 4) (Ex 4)(Ex 4)
(Ex 5) (Ex 5)
Solve the equations. Write the solutions in the form a + bi.
1. 3p2 + 7 = -9p2 + 4 2. 2x2 + 31 = 9
*3. Find the roots and give the multiplicity of each root: x3 - 7x2 + 11x - 5 = 0
4. Let f (x) = 3x + 1 and g(x) = x - 2. Find the composite function f (g(x)). (62) (62) (62)(62) (66) (66) (53) (53)
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5. Of the 220 people who came into the Italian deli on Friday, 104 bought pizza and 82 used a credit card. Half of the people who bought pizza used a credit card. What is the probability that a customer bought pizza or used a credit card?
Solve for x by completing the square.
6. x2 = 9x - 7 7. -5x - 6 = -x2
*8. Lighthouses The number of miles m a person can see on a clear day is
approximated by m = 1.2116 √ h , where h is the person’s height in feet above sea
level. If a person standing at the top of the Barnegat Lighthouse in New Jersey on a clear day can see a distance of 16 miles, what is the person’s height above sea level? 9. Error Analysis The odds in favor of an event are 9 to 2. To the nearest percent, what
is the probability of the event?
10. Multiple Choice What is P(-3) if x + 3 is a factor of 2x3 + 2x2 - 10x + 6?
A -3 B 0 C 3 D 6
11. Velocity of an Electron The velocity at which an electron orbits the nucleus of an atom is known as the Fermi-Thomas velocity and can be found using the expression Z
2
_ 3 _ 137 c, where Z is the number of protons in the nucleus and c is the speed of light. What is the Fermi-Thomas velocity for cobalt, which has 27 protons, in terms of c? Round to the nearest thousandths place.
*12. Write What is an extraneous solution? When can it occur?
13. Estimate Use what you know about the unit circle and special angles to estimate tan 43°. Do not use a calculator. Explain your method.
*14. Graphing Calculator With your calculator in a + bi mode, find the following. a. (64 - 82i)(26 + 31i) b. -3.7i(2.25 + 4.5i)
15. Electrical Engineering A circuit has a current of 5 amps, and a second circuit has a current of (14 - 9i) amps. Find the sum of the currents.
16. Temporary Employees A department store is planning to hire up to 24 temporary employees for a tent sale. Experienced workers will be paid $20 per hour and inexperienced workers will be paid $15 per hour. The company can pay up to $400 per hour for the temporary employees. Experienced workers produce 1.5 times more profit than inexperienced workers. How many of each type of worker should be hired to maximize the company’s profits?
*17. Multi-Step The product of a number and its multiplicative inverse is 1. a. State the multiplicative inverse of 2 + 3i.
b. Simplify the multiplicative inverse.
c. Verify Show that your answer in part b is the multiplicative inverse by multiplying it by 2 + 3i. (60) (60) (58) (58) (58)(58) (70) (70) (55) (55) (61) (61) (59) (59) (70) (70) (63) (63) (69) (69) (62) (62) (54) (54) (69) (69)
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Lesson 70 501
Factor each expression completely.
18. 8c3 + 343 19. -5z3 + 320
20. Generalize How does the Rational Root Theorem make factoring polynomials easier? What must be true about a polynomial before it used?
*21. Wheelchair Accessibility A contractor designed the wheelchair ramp shown in the diagram. The ramp must satisfy a building code that specifies 4.7° as the maximum angle of elevation, labeled θ in the diagram. Does the design satisfy the building code? Justify your answer.
4 ft 50 ft
θ
22. Find an equation for the inverse of y = 5x3 - 20. Determine whether each relation
is a function.
23. Verify A student used factoring to find that the solutions of 4x2 - 11x - 20 = 0
are -1.25 and 4. Show how to verify these solutions with the Quadratic Formula.
24. Maximize P = 4.5x + 2.4y for the constraints
y ≥ 0 x ≥ 0 y ≥ x y ≤ -2x + 9 25. Error Analysis A student converted the exponential equation 4 -2
= _ 1
16 to logarithmic
form by writing log4(-2) = _ 161 . What is the error? Write the correct logarithmic
equation.
26. Geometry Each side of the regular hexagon shown is 7 cm. Find the length of the diagonal AE . Round your answer to the −−
nearest tenth. (The measure of one interior angle of an n-gon is _ (n - 2)180˚n .)
27. Write Examine the formula for the probability of two inclusive events. Explain why P(A and B) is subtracted from the formula.
P(A or B)= P(A)+ P(B)- P(A and B)
28. Multi-Step A clothing store has a sale in which all the shirts are 20% off. On the last day of the sale, all shirts have an additional 25% off. Find a function describing how much a shirt with price p will cost on the last day. Solve the function.
*29. Probability A number cube is rolled twice. What is the probability that the number facing up on each roll shows a solution of √ 2x - 8 = x - 4?
*30. Multiple Choice A bag of grapes contains 9 green grapes and 15 red grapes. If 4 grapes are chosen at random, without replacement, what is the approximate probability that the first two grapes are green and the last two grapes are red?
A 4.6% B 5.9% C 7.1% D 13% (61) (61) (61)(61) (66) (66) (67) (67) (50) (50) (65) (65) (54) (54) (64) (64) A F C D E B A F C D E B (52) (52) (60) (60) (53) (53) (70) (70) (68) (68)
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