Algebra Problems of the Day (Algebra Regents, August 2024 Part III)

This exam was adminstered in August 2024 .

August 2024 Algebra, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.

31. The owner of an ice cream stand kept track of the number of ice cream cones that were sold each day of the first week in June. She compared the ice cream sales to the average daily temperature. The data are shown in the table below.

State the linear regression equation for these data, rounding all values to the nearest hundredth.

State the correlation coefficient, to the nearest hundredth, for the line of best fit for these data.

State what this correlation coefficient indicates about the linear fit of the data.

Answer:

You have to perform a Linear Regression in your calculation. Put the data from the table into two lists (L1 and L2).

After rounding the values in your calculator, you will get the equation y = 15.13x - 959.63.

The correlation coefficient, r, is 0.99.

This correlation coefficient indicates that there is a strong postivie correlation in the data.

32. Graph the system of inequalities on the set of axes below:

y > 3x - 4
x + 2y <6
Label the solution set S.
Is the point (2,2) a solution to the system? Justify your answer.

Answer:

Remember that > means a broken line with shading about the line and that < means a solid line with shading below that line.

You can use your calculator to graph the lines and use the Table of Values. Remember that the first line must be broken or dotted.

Rewrite the second equation as y < -1/2 x + 3.

Your graph will look like this:

The point (2,2) is NOT is the solution set of the system of inequalities because it is on the broken line. The points on the broken line are NOT part of the solution.

Note: your final answer will depend upon the graph you draw. If you have a graphing error, then the answer to the final part of the question MUST match your graph.

33. An object is launched upward at 64 feet per second from a platform 80 feet above the ground. The function s(t) models the height of the object t seconds after launch.

If s(t) = -16t² + 64t + 80, state the vertex of s(t), and explain in detail what each coordinate means in the context of the problem.

After the object is launched, how many seconds does it take for the object to hit the ground? Justify your answer.

Answer:

This is a problem of gravity, which creates an upside-down parabola when graphed. The vertex is a point on the Axis of Symmetry, which can be found using x = -b/(2a).

The Axis of Symmetry is x = -64/(2(-16)) = 2.

Calculate s(2) = -16(2)² + 64(2) + 80 = 144. The coordinates of the vertex are (2,144).

In the context of this problem, that point says that at two seconds the object reaches its maximum height, which is 144 feet above the ground.

The object will hit the ground at when -16t² + 64t + 80 = 0.

-16t² + 64t + 80 = 0
t² - 4t - 5 = 0
(t - 5)(t + 1) = 0
t - 5 = 0 or t + 1 = 0
t = 5 or t = -1

Discard the negative answer because time cannot be negative. The object hits the ground at t = 5 seconds.

34. Solve the system of equations algebraically for all values of x and y.

y = x² + 4x - 1
y = 2x + 7

Answer:

Set the two expressions equal to each other. Solve the quadratic equation. Then solve for y.

x² + 4x - 1 = 2x + 7
x² + 2x - 8 = 0
(x + 4)(x - 2) = 0
x + 4 = 0 or x - 2 = 0
x = -4 or x = 2

If x = -4, y = (-4)² + 4(-4) - 1 = -1.

If x = 2, y = (2)² + 4(2) - 1 = 11.

End of Part III

How did you do?

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