Question

3d objects: cones and spheres
measurement – lesson 8
assignmen

1. calculate the surface area and volume of each 3d object below. where necessary round each answer to the nearest tenth.

a.
b.
c.

Explanation:

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Part a: Rectangular Prism

Step1: Define surface area formula

Surface Area (SA) of a rectangular prism:
$$SA=2(lw + lh + wh)$$
where $l=10\ \text{ft}$, $w=7\ \text{ft}$, $h=16\ \text{ft}$

Step2: Substitute values for SA

$$SA=2((10 \times 7)+(10 \times 16)+(7 \times 16))$$
$$SA=2(70 + 160 + 112)$$
$$SA=2(342)=684.0\ \text{sq ft}$$

Step3: Define volume formula

Volume (V) of a rectangular prism:
$$V=lwh$$

Step4: Substitute values for V

$$V=10 \times 7 \times 16$$
$$V=1120.0\ \text{cu ft}$$

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Part b: Triangular Prism

Step1: Calculate triangle base area

Area of triangular base:
$$B=\frac{1}{2} \times 14 \times 10.5=73.5\ \text{sq m}$$

Step2: Calculate surface area

SA = 2(triangle area) + sum of rectangular faces:
$$SA=2(73.5)+(14 \times 17)+(10.5 \times 17)+(17.5 \times 17)$$
$$SA=147 + 238 + 178.5 + 297.5$$
$$SA=861.0\ \text{sq m}$$

Step3: Calculate volume

Volume of triangular prism:
$$V=B \times \text{length}$$
$$V=73.5 \times 17$$
$$V=1249.5\ \text{cu m}$$

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Part c: Cylinder with inner cone (note: the question asks for the 3D object; assuming we calculate the cylinder's surface area and the volume of the cylinder minus the cone, as the cone is hollowed out)

Given: Cylinder diameter = 10 m, so radius $r=5\ \text{m}$, height $h=13\ \text{m}$

Step1: Calculate cylinder total SA

Total Surface Area of cylinder:
$$SA=2\pi r^2 + 2\pi rh$$
$$SA=2\pi(5)^2 + 2\pi(5)(13)$$
$$SA=50\pi + 130\pi=180\pi \approx 565.5\ \text{sq m}$$

Step2: Calculate cylinder volume

$$V_{\text{cylinder}}=\pi r^2 h$$
$$V_{\text{cylinder}}=\pi(5)^2(13)=325\pi \approx 1021.0\ \text{cu m}$$

Step3: Calculate cone volume

Volume of cone (same r, h as cylinder):
$$V_{\text{cone}}=\frac{1}{3}\pi r^2 h$$
$$V_{\text{cone}}=\frac{1}{3}(325\pi) \approx 340.3\ \text{cu m}$$

Step4: Calculate hollow volume

$$V_{\text{hollow}}=V_{\text{cylinder}} - V_{\text{cone}}$$
$$V_{\text{hollow}} \approx 1021.0 - 340.3=680.7\ \text{cu m}$$

Answer:

a. Surface Area: $684.0$ square feet, Volume: $1120.0$ cubic feet
b. Surface Area: $861.0$ square meters, Volume: $1249.5$ cubic meters
c. Total Surface Area (cylinder): $565.5$ square meters, Hollow Volume: $680.7$ cubic meters