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Desi is looking at his calculator, which is shaped like a rectangular prism, and estimates it is about 3 in. wide, 4 in. tall, and [tex]$\frac{1}{2}$[/tex] in. thick. He wonders how many of these calculators he could fit in the trunk of his car, which is also roughly shaped like a rectangular prism, 4 ft wide, 3 ft deep, and 2 ft tall. His reasoning, shown here, contains an error.

- The calculator has a volume of about [tex]$60 \text{ in}^3$[/tex], or about [tex]$6 \times 10^1 \text{ in}^3$[/tex].
- The trunk of his car has a volume of about [tex][tex]$41,472 \text{ in}^3$[/tex][/tex], or about [tex]$4 \times 10^4 \text{ in}^3$[/tex].
- [tex]$\frac{4 \times 10^4}{6 \times 10^1} = \frac{2}{3} \times 10^3$[/tex], so 667 calculators would fit in the trunk of his car.

What is Desi's error?

A. The value 60 should have been rounded to [tex][tex]$6 \times 10^0$[/tex][/tex].
B. The value 41,472 should have been rounded to [tex]$4 \times 10^5$[/tex].
C. The volume of the car's trunk is not about [tex]$41,472 \text{ in}^3$[/tex].
D. The volume of the calculator is not about [tex][tex]$60 \text{ in}^3$[/tex][/tex].



Answer :

GinnyAnswer
Desi's error lies in the volume calculations of the calculator and the car's trunk. Here's a step-by-step detailed solution to identify the correct volumes and the mistakes Desi made:

### Step 1: Volume of Calculator
The calculator dimensions are provided:
- Width: [tex]\(3\)[/tex] inches
- Height: [tex]\(4\)[/tex] inches
- Thickness: [tex]\(0.5\)[/tex] inch

The volume [tex]\(V\)[/tex] of a rectangular prism is calculated as:
[tex]\[ V = \text{Width} \times \text{Height} \times \text{Thickness} \][/tex]
Substituting the values:
[tex]\[ V_{\text{calculator}} = 3 \text{ in} \times 4 \text{ in} \times 0.5 \text{ in} = 6 \text{ in}^3 \][/tex]

Desi incorrectly estimated the volume to be [tex]\(60 \text{ in}^3\)[/tex]. The correct volume is [tex]\(6 \text{ in}^3\)[/tex].

### Step 2: Volume of Trunk
The trunk dimensions are provided in feet, so we need to convert them to inches, since 1 foot equals 12 inches:
- Width: [tex]\(4\)[/tex] feet [tex]\(12\)[/tex] inches/foot = [tex]\(48\)[/tex] inches
- Depth: [tex]\(3\)[/tex] feet [tex]\(12\)[/tex] inches/foot = [tex]\(36\)[/tex] inches
- Height: [tex]\(2\)[/tex] feet * [tex]\(12\)[/tex] inches/foot = [tex]\(24\)[/tex] inches

Now, we calculate the volume:
[tex]\[ V = \text{Width} \times \text{Depth} \times \text{Height} \][/tex]
Substituting the values:
[tex]\[ V_{\text{trunk}} = 48 \text{ in} \times 36 \text{ in} \times 24 \text{ in} = 41472 \text{ in}^3 \][/tex]

Desi's estimated trunk volume is correctly noted as [tex]\(41,472 \text{ in}^3\)[/tex].

### Step 3: Number of Calculators that Fit in the Trunk
To find out how many calculators can fit in the trunk, we divide the volume of the trunk by the volume of a calculator:
[tex]\[ \text{Number of calculators} = \frac{\text{Volume of trunk}}{\text{Volume of calculator}} \][/tex]
Substituting the correct values:
[tex]\[ \text{Number of calculators} = \frac{41472 \text{ in}^3}{6 \text{ in}^3} = 6912 \][/tex]

Desi made a calculation error in using the calculator's volume:
[tex]\[ \frac{4 \times 10^4}{6 \times 10^1} \][/tex]
This error led him to calculate that only 667 calculators would fit. Using the correct volume of the calculator yields:
[tex]\[ \frac{41472 \text{ in}^3}{6 \text{ in}^3} = 6912 \][/tex]

### Conclusion
Desi's primary error was incorrectly estimating the volume of the calculator as [tex]\(60 \text{ in}^3\)[/tex]. The correct volume of the calculator is [tex]\(6 \text{ in}^3\)[/tex]. Thus, the correct answer is:

The volume of the calculator is not about [tex]$60 \text{ in}^3$[/tex]. This miscalculation led to the incorrect number of calculators fitting in the trunk.

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