Question

1. this trapezoidal prism is sliced with a plane parallel to the base. what is the area of the cross - section?
2. this cylinder is sliced with a plane perpendicular to the base. what is the area of the cross - section?
3. this pyramid is sliced with a plane perpendicular to the base through the center of the base. what is the area of the cross - section?
4. this cylinder is sliced with a plane parallel to the base. what is the area of the cross - section?

Explanation:

Problem 5:

Step1: Identify cross-section shape

A plane parallel to the base of a trapezoidal prism creates a cross-section congruent to the trapezoidal base. The area of a trapezoid is $\frac{1}{2}(a+b)h$, where $a$ and $b$ are the parallel sides, $h$ is the height of the trapezoid.

Step2: Substitute values

$a=5\ \text{ft}$, $b=11\ \text{ft}$, $h=9.4\ \text{ft}$
$\text{Area} = \frac{1}{2}(5+11) \times 9.4$

Step3: Calculate the result

$\frac{1}{2} \times 16 \times 9.4 = 8 \times 9.4 = 75.2$

Problem 6:

Step1: Identify cross-section shape

A plane perpendicular to the base of a cylinder creates a rectangular cross-section. The sides are the height of the cylinder and the diameter of the base.

Step2: Find diameter of base

Diameter $d = 2 \times 14 = 28\ \text{in}$, height $h=32\ \text{in}$

Step3: Calculate rectangle area

$\text{Area} = d \times h = 28 \times 32 = 896$

Problem 7:

Step1: Identify cross-section property

A plane perpendicular to the base through its center creates a triangle cross-section. The base of this triangle is half the base of the original triangular base, and the height is the same as the height of the original triangular base. First, find the area of the original base, then take $\frac{1}{4}$ (since both base and height of the cross-section triangle are half of the original, area scales by $\frac{1}{2} \times \frac{1}{2}$).

Step2: Calculate original base area

Original base area: $\frac{1}{2} \times 11 \times 10 = 55\ \text{in}^2$

Step3: Find cross-section area

$\text{Cross-section Area} = 55 \times \frac{1}{4} = 13.75$

Problem 8:

Step1: Identify cross-section shape

A plane parallel to the base of a cylinder creates a circular cross-section congruent to the base. Area of a circle is $\pi r^2$.

Step2: Substitute radius value

$r=8\ \text{mm}$
$\text{Area} = \pi \times 8^2$

Step3: Calculate the result

$\text{Area} = 64\pi \approx 200.96$

Answer:

1. $75.2\ \text{square feet}$
2. $896\ \text{square inches}$
3. $13.75\ \text{square inches}$
4. $64\pi \approx 200.96\ \text{square millimeters}$