Mathematical Reasoning Geometry and Measurement Flashcards

A frustum (truncated cone) has a bottom base radius of 9 cm, a top base radius of 4 cm, and a slant height of 13 cm. What is the lateral surface area of the frustum?

Answer: 117π cm²

The lateral surface area of a frustum is π(r₁ + r₂) × slant height = π(9 + 4) × 13 = π × 13 × 13 = 169π cm². Wait — re-checking: π(r₁ + r₂) × l = π(9 + 4)(13) = 169π. That gives 169π. But the correct formula is π(R + r)l = π(9+4)(13) = 169π cm². The answer is 169π cm².

Two similar pyramids have surface areas of 144 cm² and 256 cm² respectively. If the volume of the smaller pyramid is 216 cm³, what is the volume of the larger pyramid?

Answer: 512 cm³

The ratio of surface areas is 144:256 = 9:16, so the linear scale factor is √(9/16) = 3/4. The ratio of volumes equals the cube of the linear scale factor: (3/4)³ = 27/64. Setting up the proportion: 216/V = 27/64, so V = 216 × 64/27 = 512 cm³.

A circle is inscribed in a square with a side length of 10 cm. A point is chosen at random inside the square. What is the probability (to the nearest hundredth) that the point lies OUTSIDE the circle but INSIDE the square?

Answer: 0.21

The inscribed circle has radius 5 cm (half the side length). Area of square = 10² = 100 cm². Area of circle = π(5²) = 25π ≈ 78.54 cm². Area outside the circle but inside the square = 100 − 78.54 = 21.46 cm². Probability = 21.46/100 ≈ 0.21.

In a coordinate plane, triangle ABC has vertices at A(−3, 1), B(5, 1), and C(1, 7). What is the equation of the median from vertex C to side AB?

Answer: x = 1

The median from C goes to the midpoint of AB. The midpoint of AB is ((−3+5)/2, (1+1)/2) = (1, 1). The median connects C(1, 7) to the midpoint (1, 1). Both points have x-coordinate 1, so the median is the vertical line x = 1.

A cylindrical water tank has a diameter of 8 feet and a height of 15 feet. Water is pumped out at a rate of 4π cubic feet per minute. How many minutes will it take to lower the water level by exactly 3 feet?

Answer: 12 minutes

The volume of water corresponding to a 3-foot drop in a cylinder with radius 4 feet is: V = π r² h = π(4²)(3) = 48π cubic feet. At a rate of 4π cubic feet per minute, the time required is 48π ÷ 4π = 12 minutes.

The coordinates of the endpoints of a diameter of a circle are (−2, 3) and (6, −5). What is the equation of this circle?

Answer: (x − 2)² + (y + 1)² = 32

The center is the midpoint of the diameter: ((−2+6)/2, (3+(−5))/2) = (2, −1). The radius is half the diameter length. Diameter = √[(6−(−2))² + (−5−3)²] = √[64 + 64] = √128 = 8√2. Radius = 4√2, so r² = 32. The equation is (x − 2)² + (y + 1)² = 32.