Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.
After a brief hiatus, the Algebra 2 Problems of the Day are back. Hopefully, daily.

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January 2019, Part III

All Questions in Part I are worth 4 credits. Partial credit can be earned.


35.A radio station claims to its advertisers that the mean number of minutes commuters listen to the station is 30. The station conducted a survey of 500 of their listeners who commute. The sample statistics are shown below.

A simulation was run 1000 times based upon the results of the survey. The results of the simulation appear below.

Based on the simulation results, is the claim that commuters listen to the station on average 30 minutes plausible? Explain your response including an interval containing the middle 95% of the data, rounded to the nearest hundredth.

Answer:
The mean is 29.101. The middle 95% is two standard deviations above and below the mean.
29.101 - 2(0.934) = 27.233
29.101 + 2(0.934) = 30.969
The interval is 27.23 - 30.97
Yes, 30 falls within the interval so the claim is plausible.

36.Solve the given equation algebraically for all values of x.

3 * sqrt (x) - 2x = -5

Answer:
Isolate the radical term by adding 2x to both sides. Then square both sides and solve the quadratic equation.
Discard any negative results because of the radical. (x cannot be less than 0.)
Check for extraneous results

3 sqrt (x) - 2x = -5
3 sqrt (x) = 2x - 5
(3 sqrt (x) ) ² = (2x - 5) ²
9x = 4x² - 20x + 25
0 = 4x² - 29x + 25
0 = 4x² - 4x - 25x + 25
0 = (4x)(x - 1) - (25)(x - 1)
0 = (4x - 25)(x - 1)
4x - 25 = 0 or x - 1 = 0
4x = 25 or x = 1
x = 6.25 or x = 1

Check: 3 (6.25)^.5 - 2(6.25) = 3(2.5) - 12.5 = 7.5 - 12.5 = -5 (check)
3 (1)^.5 - 2(.5) = 3(1) - 1 = 3 - 1 = 2 =/= -5 (reject)

x = 6.25.



Comments and questions welcome.

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