Michael C.

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Geometry

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Question:

A 6 ft tall student casts a shadow 2 ft long. At the same time, a nearby building casts a 12 ft. shadow. How tall is the building?

Michael C.

Answer:

This can best be answered by using similar triangles. Drawing triangles to represent the student and the student's shadow, and the building and its shadow will show how similar triangle properties can be applied. The corresponding sides of the triangles are proportional. This means that 6/2 = x/12. Solving for x, we get x=36, meaning the height of the building is 36 feet.

Calculus

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Question:

How can you determine if the graph of a function is concave up or concave down?

Michael C.

Answer:

There are many ways to tell if a graph of a function is concave up or concave down. The easiest way is to graph the function and then look at the graph. If the slope of the curve is increasing over an interval, then the graph is concave up on that interval. If the slope is decreasing, then it is concave down on that interval. If a graph is not an option, then one can use the second derivative test. This test states that if the second derivative is positive, signifying that the slope is constantly increasing, then the graph is concave up for the entire interval where the second derivative is positive. Likewise, the graph is concave down for an interval where the second derivative is negative. Using the second derivative of a function can help you determine the concavity and change in concavity (inflection points) for every interval of the graph.

Algebra

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Question:

When finding the extrema of a parabola, what are the steps you should take, and why?

Michael C.

Answer:

The extrema of a parabola is the highest or lowest point on the graph. (The lowest point for a parabola opening up, and the highest point for a parabola opening down.) Since the graph of a parabola is symmetric around the axis of symmetry, you should find the location of this axis. With the equation in standard form, plug the coefficients into the axis of symmetry equation. (x=b/2a). This gives you the location of the axis of symmetry, and the x-coordinate of the extrema. Plugging this value into the original equation, give you the y-coordinate. After you have both, you have the location of the highest or lowest point on the parabola.

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