You measure the dimensions of a rectangular block of aluminum to determine its volume. It has the following dimensions.
length = 10.27 cm; width = 4.92 cm; height = 1.53 cm
Assuming that each of these measurements is accurate to the nearest 0.01 cm, find the volume of the block and the fractional error on its volume due to measurement errors.
I keep getting .007 cm^3 what am I doing wrong?
To find the volume of the rectangular block, you need to multiply its three dimensions (length, width, and height) together.
In this case, the measurements are given to the nearest 0.01 cm, so we can assume a possible error of +/- 0.005 cm for each measurement. This means the actual length could be as small as 10.265 cm or as large as 10.275 cm, the actual width could be as small as 4.915 cm or as large as 4.925 cm, and the actual height could be as small as 1.525 cm or as large as 1.535 cm.
To account for the potential errors, we will calculate the volume using the maximum values and the minimum values for each dimension.
Using the maximum values:
Length = 10.275 cm
Width = 4.925 cm
Height = 1.535 cm
Volume (maximum) = Length x Width x Height = 10.275 cm x 4.925 cm x 1.535 cm = 79.945 cm³
Using the minimum values:
Length = 10.265 cm
Width = 4.915 cm
Height = 1.525 cm
Volume (minimum) = Length x Width x Height = 10.265 cm x 4.915 cm x 1.525 cm = 79.850 cm³
So, the volume of the rectangular block lies between 79.850 cm³ and 79.945 cm³.
To calculate the fractional error on the volume, we can subtract the minimum volume from the maximum volume and divide it by the average volume.
Average volume = (79.850 cm³ + 79.945 cm³) / 2 = 79.8975 cm³
Fractional error = (79.945 cm³ - 79.850 cm³) / 79.8975 cm³ ≈ 0.00119
Therefore, the fractional error on the volume is approximately 0.00119 or 0.119%.
Based on the calculations, the correct volume should be between 79.850 cm³ and 79.945 cm³, rather than the value of 0.007 cm³ that you obtained.
To find the volume of the rectangular block, you need to multiply its length, width, and height together. Let's calculate it step-by-step using the given measurements:
1. Length = 10.27 cm
2. Width = 4.92 cm
3. Height = 1.53 cm
Volume = Length * Width * Height
Volume = 10.27 cm * 4.92 cm * 1.53 cm
Volume = 80.033308 cm^3
Therefore, the volume of the rectangular block is approximately 80.033308 cm^3.
Now, let's calculate the fractional error on its volume due to measurement errors. The fractional error can be determined by dividing the actual error by the measured value.
In this case, the measured value is 80.033308 cm^3, and the actual error is caused by the rounding of measurements, which is 0.01 cm for each dimension.
To determine the actual error, you need to calculate the volume using the highest possible value for each dimension (adding 0.01 cm to each dimension) and the lowest possible value for each dimension (subtracting 0.01 cm from each dimension).
Highest volume (adding 0.01 cm to each dimension):
Length = 10.28 cm
Width = 4.93 cm
Height = 1.54 cm
Volume_high = 10.28 cm * 4.93 cm * 1.54 cm
Volume_high = 80.940904 cm^3
Lowest volume (subtracting 0.01 cm from each dimension):
Length = 10.26 cm
Width = 4.91 cm
Height = 1.52 cm
Volume_low = 10.26 cm * 4.91 cm * 1.52 cm
Volume_low = 79.126192 cm^3
Now, calculate the actual error:
Actual error = (Volume_high - Volume_low) / 2
Actual error = (80.940904 cm^3 - 79.126192 cm^3) / 2
Actual error = 1.814712 cm^3 / 2
Actual error = 0.907356 cm^3
Finally, calculate the fractional error:
Fractional error = Actual error / Measured value
Fractional error = 0.907356 cm^3 / 80.033308 cm^3
Fractional error = 0.01134
Therefore, the fractional error on the volume of the block due to measurement errors is approximately 0.01134 or 1.134%.
Hence, your previous answer of 0.007 cm^3 was incorrect. The correct volume of the block is approximately 80.033308 cm^3, and the fractional error is 1.134%.
L min. = 10.26 cm
W min. = 4.91 cm
H min. = 1.52 cm
V nom. = 10.27 * 4.92 * 1.53=77.308 cm^3
V min. = 10.26 * 4.91 * 1.52=76.728cm^3
Error = 77.308 - 76.728 = 0.5796 cm^3
NOTE: The fractional error has no units;
they cancel.
Fractional Error = 0.58cm^3/77.308cm^3 =
0.0075