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.

More Algebra 2 problems.

January 2019, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.


7. Tides are a periodic rise and fall of ocean water. On a typical day at a seaport, to predict the time of the next high tide, the most important value to have would be the

(1) time between consecutive low tides
(2) time when the tide height is 20 feet
(3) average depth of water over a 24-hour period
(4) difference between the water heights at low and high tide

Answer: (1) time between consecutive low tides
If the tides are periodic, then knowing the consecutive low tides will tell you when the next high tide is because it will be at the midpoint of those two times.
The water height by itself will not let you predict the next high tide.

8. An estimate of the number of milligrams of a medication in the bloodstream t hours after 400 mg has been taken can be modeled by the function below.

I(t) = 0.5t⁴ + 3.45t³ - 96.65t² + 347.7t, where 0 ≤ t ≤ 6 Over what time interval does the amount of medication in the bloodstream strictly increase?

(1) 0 to 2 hours
(2) 0 to 3 hours
(3) 2 to 6 hours
(4) 3 to 6 hours

Answer: (1) 0 to 2 hours
If you graph the function, you will see that it it's rising from 0 to 2. It reaches a local maximum point at approximately x = 2.15. The graph decreases between x = 2.15 and x = 6.

9. Which representation of a quadratic has imaginary roots?

Answer: (4) 2x² + 32 = 0
If the quadratic intersects or crosses the x-axis then it does not have imaginary roots.
In other words, if there is some value of x which makes y equal to 0, it has a real root.
Choice (1) has (-2.0, 0) and Choice (3) has (3, 0), so they both can be eliminated.

The equations in (2) and (4) have imaginary roots, if there are no real solutions that make the equation true.
If 2(x + 3)² = 64
then (x + 3)² = 32
and x + 3 = SQRT(32)
So x = -3 + SQRT(32), which is a real, irrational value.

If 2x² + 32 = 0
then 2x² = -32
and x² = -16.
This means x = SQRT(-16), which is not a real number.



Comments and questions welcome.

More Algebra 2 problems.