Question

1. an aquarium weighs 22.5 pounds when empty. the aquarium is 30 in. long, 14 in. wide, and is filled with water to a depth of 18 in. given 1 in³ of water weighs 0.036 pounds, determine the weight of the aquarium when it is full of water. provide a diagram.
2. the rectangular prism below has a volume of 432 ft³. determine the height of the prism in inches.
3. a rectangular prism with a square base with sides of 5 cm each has a surface area of 170 cm². find the height of the rectangular prism. provide a diagram.

Explanation:

Problem 6

Step 1: Calculate the volume of water

The aquarium is a rectangular prism, so the volume \( V \) of water is given by the formula \( V = l \times w \times h \), where \( l = 30 \) in, \( w = 14 \) in, and \( h = 18 \) in.
\[
V = 30 \times 14 \times 18
\]
\[
V = 30 \times 252
\]
\[
V = 7560 \text{ in}^3
\]

Step 2: Calculate the weight of the water

Given that 1 in³ of water weighs 0.036 pounds, the weight of the water \( W_{water} \) is the volume of water times the weight per cubic inch.
\[
W_{water} = 7560 \times 0.036
\]
\[
W_{water} = 272.16 \text{ pounds}
\]

Step 3: Calculate the total weight of the aquarium

The total weight \( W_{total} \) is the weight of the empty aquarium plus the weight of the water. The empty aquarium weighs 22.5 pounds.
\[
W_{total} = 22.5 + 272.16
\]
\[
W_{total} = 294.66 \text{ pounds}
\]

(Diagram: A rectangular prism labeled with length 30 in, width 14 in, height 18 in, representing the aquarium filled with water. The empty weight is 22.5 lb, and the water weight is calculated as above.)

Step 1: Convert the volume to cubic inches

The volume of the rectangular prism is given in cubic feet (\( 432 \text{ ft}^3 \)). We know that \( 1 \text{ ft} = 12 \text{ in} \), so \( 1 \text{ ft}^3 = 12^3 = 1728 \text{ in}^3 \).
\[
V_{in^3} = 432 \times 1728
\]
Wait, no—actually, the formula for the volume of a rectangular prism is \( V = l \times w \times h \). We know \( l = 16 \) ft, \( w = 3 \) ft, and we need to find \( h \) in inches. First, let's find \( h \) in feet using \( V = l \times w \times h \).
\[
432 = 16 \times 3 \times h
\]
\[
432 = 48 \times h
\]
\[
h = \frac{432}{48}
\]
\[
h = 9 \text{ ft}
\]

Step 2: Convert height from feet to inches

Since \( 1 \text{ ft} = 12 \text{ in} \),
\[
h_{in} = 9 \times 12
\]
\[
h_{in} = 108 \text{ inches}
\]

Step 1: Recall the surface area formula for a rectangular prism with a square base

The surface area \( SA \) of a rectangular prism with a square base (side \( s \)) and height \( h \) is \( SA = 2s^2 + 4sh \). Here, \( s = 5 \) cm and \( SA = 170 \) cm².

Step 2: Substitute the known values into the formula

\[
170 = 2(5)^2 + 4(5)h
\]
\[
170 = 2(25) + 20h
\]
\[
170 = 50 + 20h
\]

Step 3: Solve for \( h \)

Subtract 50 from both sides:
\[
170 - 50 = 20h
\]
\[
120 = 20h
\]
Divide both sides by 20:
\[
h = \frac{120}{20}
\]
\[
h = 6 \text{ cm}
\]

(Diagram: A rectangular prism with a square base (side 5 cm) and height \( h \), labeled with surface area 170 cm². The base is a square, and the four lateral faces are rectangles with height \( h \) and width 5 cm.)

Answer:

The weight of the aquarium when full of water is \( \boldsymbol{294.66} \) pounds.

Problem 7