Blocks of stone are cut to a nominal size of 60cm x 30cm x 15cm, where measurements are correct to the nearest centimeter. Find the range of values of the volumes of a block giving your answer in the form V± pV, where V is the nominal volume and p is the percentage.
What's the answer please?
60cm x 30cm x 15cm = 27,000
59.5 * 29.5 * 14.5 = 25,451
60.5 * 30.5 * 15.5 = 28,601
28601 - 25451 = 3,150
3150/2 = 1575
1575/27000 = .0583 = 5.8%
27,000 +/- 5.8 % * 27,000
or, note that
.5/15 = 0.033333 = 1/30
.5/30 = 1/60
.5/60 = 1/120
1/30 + 1/60 + 1/120 = 7/120 = 0.0583 as above
when multiplying factors, add their relative errors
To find the range of values for the volume of the block, we first need to calculate the nominal volume. The nominal volume is calculated by multiplying together the three dimensions of the block: length, width, and height.
The nominal volume (V) of the block is given as:
V = Length x Width x Height
Plugging in the values, we get:
V = 60cm x 30cm x 15cm = 27,000 cm³
Now, we need to calculate the possible variations in the dimensions, taking into account that measurements are correct to the nearest centimeter.
The variation can be either in the positive or negative direction. Since the measurements are accurate to the nearest centimeter, the positive variation would be when we increase each dimension by 0.5 centimeters. Similarly, the negative variation is when we decrease each dimension by 0.5 centimeters.
Hence, the positive variation is 0.5 cm and the negative variation is -0.5 cm.
Now, we can calculate the maximum and minimum volumes:
Maximum volume (Vmax):
Vmax = (Length + positive variation) x (Width + positive variation) x (Height + positive variation)
Plugging in the values, we get:
Vmax = (60cm + 0.5cm) x (30cm + 0.5cm) x (15cm + 0.5cm)
= (60.5cm) x (30.5cm) x (15.5cm)
= 28,507.375 cm³
Minimum volume (Vmin):
Vmin = (Length + negative variation) x (Width + negative variation) x (Height + negative variation)
Plugging in the values, we get:
Vmin = (60cm - 0.5cm) x (30cm - 0.5cm) x (15cm - 0.5cm)
= (59.5cm) x (29.5cm) x (14.5cm)
= 24,732.875 cm³
Finally, we can find the range of values by subtracting Vmin from V and Vmax from V:
Range = Vmax - Vmin = 28,507.375 cm³ - 24,732.875 cm³ = 3,774.5 cm³
Expressing the range in the form V ± pV, where V is the nominal volume and p is the percentage:
Range = 27,000 cm³ ± (3,774.5 cm³ / 27,000 cm³) * 100
Simplifying further, we get:
Range = 27,000 cm³ ± 14%
Therefore, the range of values for the volumes of the block is 27,000 cm³ ± 14% (approximately).