The boy having a gold coin of diameter 2cm and thickess 2mm to a goldsmith for melting and drawing into a wire of diameter 1mm . the gold smith gave back a wire of gold of length 76cm with diameter 1mm. if the wastage allowed in melting could be a maximum of 5% of the total volume , does the godsmith man was honest or not
If the smith was honest, you want
pi (.05)^2 * 76 >= 0.95 pi (1)^2 * .2
.19 >= .19
Looks like the smith was honest.
from where .95 comes
5% waste means at least 95% of the metal must be left after working on it.
To determine whether the goldsmith was honest or not, we need to compare the volume of gold in the initial coin with the volume of gold in the final wire, taking into account the allowed wastage.
Let's calculate the volumes involved:
Volume of the initial gold coin:
V1 = π * (r1^2) * h1
= π * (1cm^2) * 0.2cm
= 0.2π cm^3
Volume of the final gold wire:
V2 = π * (r2^2) * h2
= π * (0.05cm^2) * 76cm
= 3.8π cm^3
Now, let's calculate the maximum allowable wastage:
Maximum allowable wastage = 5% of the total volume
= 0.05 * (V1 + V2)
To determine if the goldsmith was honest or not, we need to check if the actual wastage is less than or equal to the maximum allowable wastage.
Actual wastage = V1 - V2
If the actual wastage is within the allowed range, the goldsmith can be considered honest. If the actual wastage exceeds the allowed range, the goldsmith may not have been honest.
Now, let's calculate the values:
Actual wastage = 0.2π - 3.8π
= -3.6π cm^3
Maximum allowable wastage = 0.05 * (V1 + V2)
= 0.05 * (0.2π + 3.8π)
= 0.05 * 4π
= 0.2π cm^3
Since the actual wastage of -3.6π cm^3 is higher than the maximum allowable wastage of 0.2π cm^3, it indicates that the goldsmith may not have been honest.