Friday, November 19, 2021

Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2012

Part III: Each correct answer will receive 4 credits. Partial credit is available.

36. The diagram below shows the plans for a cell phone tower. A guy wire attached to the top of the tower makes an angle of 65 degrees with the ground. From a point on the ground 100 feet from the end of the guy wire, the angle of elevation to the top of the tower is 32 degrees. Find the height of the tower, to the nearest foot.

We are concerned with the two smaller triangles: the one with the guy wire and the 62-degree angle, and the one with the guy wire and the tower. You need to use the Law of Sines to find the length of the guy wire. You can then use that as the hypotenuse of the right triangle to find the height of the tower.

The Exterior Angle Theorem tells us that the top angle in the left side triangle is 65 - 32 = 33 degrees. So:

sin 32 / x = sin 33 / 100

x = 100 * sin 32 / sin 33 = 97.297 ...

We have the hypotenuse of the right side triangle and we need the opposite side, so use sine.

sin 65 = x / 162.116

x = 97.297 * sin 65 = 88.18

The height of the tower is 88 feet.

37. If log 4 x = 2.5 and log y 125 = -3/2, find the numerical value of x/y, in simplest form.

Use inverse operations.

log 4 x = 2.5

42.5 = x

45/2 = x

25 = x

x = 32

log y 125 = -3/2

y-3/2 = 125

1/y3/2 = 125

y3/2 = 1/125

y1/2 = 1/5

y = 1/25

Substitute: x/y = 32 / (1/25) = 800

38. A population of single-celled organisms was grown in a Petri dish over a period of 16 hours. The number of organisms at a given time is recorded in the table below.

Determine the exponential regression equation model for these data, rounding all values to the nearest ten-thousandth.

Using this equation, predict the number of single-celled organisms, to the nearest whole number, at the end of the 18th hour.

Put the data into lists in your graphing calculator, and run an exponential regression.

The nearest ten-thousandth means four decimal places. You should have gotten a = 27.2025 and b = 1.1509, so the equation is y = 27.2025(1.1509)x.

To predict the number of organisms, substitute 18 for x and calculate.

y = 27.2025(1.1509)18 = 341.41... = 341.

The prediction is 341 organisms.

End of Part III.

More to come. Comments and questions welcome.

More Regents problems.

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