Complex Factorization and Volume of a Sphere: Precalculus Math Test – PDF Download

Question 1: Powers of i (2 points)

Simplify.

1. \(i^{38}\) (1 point)
2. \(i^{55}\) (1 point)

Question 2: Factor a sum of squares (2 points)

Factor \(25x^{2}+4\) over the complex numbers. (1 point) Then expand your answer to verify it. (1 point)

Question 3: Solve over the complex numbers (2 points)

Solve.

1. \(x^{2}+64=0\) (1 point)
2. \(x^{2}+18=0\) (1 point)

Question 4: Fourth-degree factorization (3 points)

Factor \(x^{4}-256\) completely over the complex numbers.

Question 5: Quadratic with no real zeros (3 points)

The graph of \(y=x^{2}-6x+25\) is shown.

1. Explain from the graph why the quadratic has no real zeros. (1 point)
2. Factor \(x^{2}-6x+25\) over the complex numbers. (2 points)

Question 6: Matching cross-sections (3 points)

A hemisphere of radius \(5\) ft and a cylinder of radius \(5\) ft and height \(5\) ft with a cone removed are each cut by a plane \(3\) ft above their base.

1. Find the area of the cross-section of the hemisphere. (1 point)
2. Find the area of the cross-section of the cylinder with the cone removed. (1 point)
3. State what Cavalieri’s principle concludes if this holds at every height. (1 point)

Question 7: Volume from a cylinder and a cone (2 points)

A cylinder and a cone each have radius \(6\) in. and height \(6\) in.

1. Find the volume of the cylinder with the cone removed. (1 point)
2. Use that result to find the volume of a sphere of radius \(6\) in. (1 point)

Question 8: Spherical tank (3 points)

A spherical storage tank has a diameter of \(20\) ft.

1. Find its exact volume. (1 point)
2. Find its volume to the nearest tenth of a cubic foot. (1 point)
3. Find the capacity in gallons to the nearest gallon, using 1 cubic foot = 7.48 gallons. (1 point)

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