bell ringer: 1. a right triangle has a hypotenuse length of 20 inches and a leg with a length of 10 inches. what is the length of the other leg rounded to the nearest tenths.
today
2. find the surface area.
(there is a rectangular prism with dimensions 22 ft, 9 ft, 6.5 ft, and some handwritten notes: p=37.5, b=143, h=9, s=ph+2b, (37.5)(9)+2(143), 623.5)
3. find the lateral surface area.
(there is a cube with side length 9 m)
4. find the lateral surface area.
(there is a rectangular prism with base, dimensions 20 in, 15 in, 10 in, and handwritten note: s=ph)
5. find the surface area.
(there is a rectangular prism with dimensions 16 cm, 8 cm, 5 cm)

Question

Accepted Answer

### Problem 3: Lateral Surface Area of a Cube (9m side)
# Explanation:
## Step 1: Recall Lateral Surface Area Formula for Cube
For a cube with side length $ s $, the lateral surface area (LSA) is given by $ \text{LSA} = 4s^2 $ (since a cube has 4 lateral faces, each with area $ s^2 $).

## Step 2: Substitute $ s = 9 \, \text{m} $
Substitute $ s = 9 $ into the formula:
$ \text{LSA} = 4 \times (9)^2 $

## Step 3: Calculate $ 9^2 $ and Multiply
First, $ 9^2 = 81 $. Then, $ 4 \times 81 = 324 $.

# Answer:
The lateral surface area is $ 324 \, \text{square meters} $ (or $ 324 \, \text{m}^2 $).

### Problem 4: Lateral Surface Area of a Rectangular Prism (20 in, 15 in, 10 in)
# Explanation:
## Step 1: Recall Lateral Surface Area Formula for Rectangular Prism
For a rectangular prism with length $ l $, width $ w $, and height $ h $, the lateral surface area (LSA) is given by $ \text{LSA} = 2h(l + w) $ (or $ \text{LSA} = Ph $, where $ P $ is the perimeter of the base: $ P = 2(l + w) $).

## Step 2: Identify Dimensions
From the diagram, assume the base has length $ l = 20 \, \text{in} $, width $ w = 15 \, \text{in} $, and height $ h = 10 \, \text{in} $ (or adjust based on labeling; here, $ h = 10 $ as the vertical side).

## Step 3: Calculate Perimeter of the Base ($ P $)
$ P = 2(l + w) = 2(20 + 15) = 2(35) = 70 \, \text{in} $.

## Step 4: Calculate Lateral Surface Area ($ \text{LSA} = Ph $)
Substitute $ P = 70 $ and $ h = 10 $:
$ \text{LSA} = 70 \times 10 = 700 $.

# Answer:
The lateral surface area is $ 700 \, \text{square inches} $ (or $ 700 \, \text{in}^2 $).

### Problem 5: Surface Area of a Rectangular Prism (16 cm, 8 cm, 5 cm)
# Explanation:
## Step 1: Recall Surface Area Formula for Rectangular Prism
The total surface area (SA) of a rectangular prism is given by $ \text{SA} = 2(lw + lh + wh) $, where $ l = \text{length} $, $ w = \text{width} $, $ h = \text{height} $.

## Step 2: Identify Dimensions
From the diagram, $ l = 16 \, \text{cm} $, $ w = 8 \, \text{cm} $, $ h = 5 \, \text{cm} $.

## Step 3: Calculate Each Pair of Faces
- Area of $ lw $ faces: $ 2(lw) = 2(16 \times 8) = 2(128) = 256 $
- Area of $ lh $ faces: $ 2(lh) = 2(16 \times 5) = 2(80) = 160 $
- Area of $ wh $ faces: $ 2(wh) = 2(8 \times 5) = 2(40) = 80 $

## Step 4: Sum the Areas
Add the three results: $ 256 + 160 + 80 = 496 $.

# Answer:
The total surface area is $ 496 \, \text{square centimeters} $ (or $ 496 \, \text{cm}^2 $).

### Problem 2: Surface Area of a Rectangular Prism (22 ft, 9 ft, 6.5 ft)
# Explanation:
## Step 1: Recall Total Surface Area Formula for Rectangular Prism
The total surface area (SA) of a rectangular prism is $ \text{SA} = 2(lw + lh + wh) $, where $ l = 22 \, \text{ft} $, $ w = 9 \, \text{ft} $, $ h = 6.5 \, \text{ft} $.

## Step 2: Calculate Each Pair of Faces
- $ lw $ faces: $ 2(22 \times 9) = 2(198) = 396 $
- $ lh $ faces: $ 2(22 \times 6.5) = 2(143) = 286 $
- $ wh $ faces: $ 2(9 \times 6.5) = 2(58.5) = 117 $

## Step 3: Sum the Areas
Add the three results: $ 396 + 286 + 117 = 799 $.

# Answer:
The total surface area is $ 799 \, \text{square feet} $ (or $ 799 \, \text{ft}^2 $).

### Problem 5 (Second Prism: 16 cm, 8 cm, 5 cm) – Surface Area (Repeated for Clarity)
# Explanation:
## Step 1: Formula for Total Surface Area (TSA) of Rectangular Prism
$ \text{TSA} = 2(lw + lh + wh) $, where $ l = 16 \, \text{cm} $, $ w = 8 \, \text{cm} $, $ h = 5 \, \text{cm} $.

## Step 2: Substitute Values
$ \text{TSA} = 2[(16 \times 8) + (16 \times 5) + (8 \times 5)] $

## Step 3: Calculate Each Term Inside the Brackets
- $ 16 \times 8 = 128 $
- $ 16 \times 5 = 80 $
- $ 8 \times 5 = 40 $

## Step 4: Sum the Terms and Multiply by 2
Sum: $ 128 + 80 + 40 = 248 $. Then, $ 2 \times 248 = 496 $.

# Answer:
The total surface area is $ 496 \, \text{square centimeters} $ (or $ 496 \, \text{cm}^2 $).

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QUESTION IMAGE

Question

Explanation:

Problem 3: Lateral Surface Area of a Cube (9m side)

Step 1: Recall Lateral Surface Area Formula for Cube

Step 2: Substitute \( s = 9 \, \text{m} \)

Step 3: Calculate \( 9^2 \) and Multiply

Step 1: Recall Lateral Surface Area Formula for Rectangular Prism

Step 2: Identify Dimensions

Step 3: Calculate Perimeter of the Base (\( P \))

Step 4: Calculate Lateral Surface Area (\( \text{LSA} = Ph \))

Step 1: Recall Surface Area Formula for Rectangular Prism

Step 2: Identify Dimensions

Step 3: Calculate Each Pair of Faces

Step 4: Sum the Areas

Answer:

Problem 4: Lateral Surface Area of a Rectangular Prism (20 in, 15 in, 10 in)