Worked Examples: SAT Strategy
Example 1: Easy (Working Backward from Volume)
Question: A right circular cylinder has a volume of $72\pi$ cubic centimeters and a height of $8$ centimeters. What is the radius of the base of the cylinder?
SAT Strategy: Pull the correct formula from the Reference Sheet ($V = \pi r^2 h$), substitute the known values, and use basic algebra to isolate the unknown.
- Set up the equation: $72\pi = \pi r^2 (8)$.
- Divide both sides by $\pi$: $72 = 8r^2$.
- Divide by 8: $9 = r^2$.
- Take the square root: $r = 3$.
- Answer: 3.
Example 2: Medium (Dimensional Scaling)
Question: Cylinder A has a radius of $r$ and a height of $h$. Cylinder B has a radius of $2r$ and a height of $\frac{h}{2}$. The volume of Cylinder B is how many times the volume of Cylinder A?
SAT Strategy: Write out the volume formula for both solids using the given variables, being extremely careful to put the changed radius entirely inside parentheses before squaring it.
- Volume of Cylinder A: $V_A = \pi r^2 h$.
- Volume of Cylinder B: $V_B = \pi (2r)^2 (\frac{h}{2})$.
- Expand the squared term: $V_B = \pi (4r^2) (\frac{h}{2})$.
- Simplify the constants: $V_B = 2\pi r^2 h$.
- Compare $V_B$ to $V_A$: $2\pi r^2 h$ is exactly $2 \times (\pi r^2 h)$.
- Answer: 2.
Example 3: Hard (Algebraic Geometry)
Question: A right rectangular prism has a volume represented by the expression $x^3 - 4x$. The area of the rectangular base is represented by the expression $x^2 - 2x$. Assuming $x > 2$, which expression represents the height of the prism?
SAT Strategy: Use your AA SL factoring skills. The volume of a prism is simply $V = \text{Base Area} \times \text{height}$. Therefore, $h = \frac{V}{\text{Base Area}}$.
- Set up the division: $h = \frac{x^3 - 4x}{x^2 - 2x}$.
- Factor the numerator: $x(x^2 - 4) \implies x(x - 2)(x + 2)$.
- Factor the denominator: $x(x - 2)$.
- Cancel the common terms ($x$ and $x-2$) from the top and bottom.
- The remaining expression is $x + 2$.
- Answer: $x + 2$.
Practice Questions
Level 1: Easy (Foundational SAT)
- A right rectangular prism has a volume of $120$ cubic inches. If the length is $5$ inches and the width is $6$ inches, what is the height of the prism? (Grid-in)
- A right circular cylinder has a volume of $45\pi$. If the height of the cylinder is $5$, what is the radius?
- A) $3$
- B) $9$
- C) $15$
- D) $40$
- The total surface area of a cube is $150$ square centimeters. What is the volume of the cube, in cubic centimeters? (Grid-in)
- A sphere has a radius of $r$. If the radius of the sphere is tripled, by what factor does the volume of the sphere increase?
- A) $3$
- B) $6$
- C) $9$
- D) $27$
- A right circular cone has a volume of $24\pi$ and a base radius of $6$. What is the height of the cone? (Grid-in)
Level 2: Medium (Standard SAT)
- A rectangular prism has dimensions of $x$, $2x$, and $3x$. If the total surface area of the prism is $88$ square units, what is the value of $x$? (Grid-in)
- Solid A and Solid B are geometrically similar. The surface area of Solid A is exactly $4$ times the surface area of Solid B. The volume of Solid A is how many times the volume of Solid B?
- A) $2$
- B) $4$
- C) $8$
- D) $16$
- The volume of a pyramid is $V$. The base of the pyramid is a square with side length $s$. Which of the following equations correctly expresses the height $h$ of the pyramid in terms of $V$ and $s$?
- A) $h = \frac{3V}{s^2}$
- B) $h = \frac{V}{3s^2}$
- C) $h = \frac{\sqrt{3V}}{s}$
- D) $h = 3Vs^2$
- Cube A has a volume of $64$ cubic units. Cube B has a side length that is $2$ units longer than the side length of Cube A. What is the volume of Cube B? (Grid-in)
- A hemisphere (half of a sphere) has a volume of $18\pi$. What is the radius of the hemisphere?
- A) $3$
- B) $6$
- C) $9$
- D) $27$
Level 3: Hard (Advanced SAT)
- The volume of a right circular cylinder is $45\pi$. If the radius is halved and the height is tripled, what is the volume of the new cylinder?
- A) $33.75\pi$
- B) $67.5\pi$
- C) $135\pi$
- D) $337.5\pi$
- A right circular cone and a right circular cylinder have the exact same radius and the exact same volume. If the height of the cone is $12$, what is the height of the cylinder? (Grid-in)
- A composite solid is formed by attaching a right circular cone perfectly on top of a right circular cylinder. Both the cone and the cylinder have a radius of $3$. If the cylinder has a height of $4$ and the cone also has a height of $4$, what is the total volume of the composite solid? (Grid-in)
- A rectangular pool is $10$ meters long and $5$ meters wide. It is filled with water to a depth of $2$ meters. If water is pumped out of the pool at a constant rate of $4$ cubic meters per hour, how many hours will it take to completely empty the pool? (Grid-in)
- The surface area of a sphere is $36\pi$ square units. What is the volume of the sphere?
- A) $12\pi$
- B) $27\pi$
- C) $36\pi$
- D) $108\pi$
Level 4: Very Hard (Slightly beyond SAT)
- A right rectangular prism has a volume represented by the expression $x^3 - 9x$. The area of the rectangular base of the prism is represented by $x^2 + 3x$. If $x > 3$, which expression represents the height of the prism?
- A) $x - 3$
- B) $x + 3$
- C) $x^2 - 3x$
- D) $x(x - 3)$
- A cylinder holds $100$ mL of liquid. A geometrically similar cylinder is created such that all of its 1-dimensional measurements (radius and height) are exactly $2.5$ times larger than the original cylinder's measurements. What is the capacity of the larger cylinder, in mL? (Grid-in)
- A solid metal cone has a radius of $6$ cm and a height of $10$ cm. The cone is melted down completely and the metal is used to cast identical smaller cones, each with a radius of $2$ cm and a height of $5$ cm. How many small cones can be produced? (Grid-in)
- A cube is perfectly inscribed inside a sphere of radius $\sqrt{3}$. What is the volume of the cube?
- A) $3\sqrt{3}$
- B) $8$
- C) $12$
- D) $24\sqrt{3}$
- The volume of a sphere is $V$. If the radius of the sphere is decreased by $50%$, what is the surface area of the new sphere in terms of the original volume $V$?
- A) It cannot be determined without the numerical value of the radius.
- B) $\pi (\frac{3V}{4\pi})^{2/3}$
- C) $(\frac{9V^2}{16\pi})^{1/3}$
- D) $\pi (\frac{3V}{32\pi})^{2/3}$
🔒 Answer Key & Brief Explanations
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