Answer
To meet the order, the plant manager should produce 11,804 rods in total.
Work Step by Step
Want to find $P(24.9 < x < 25.1)$
i) Convert 24.9 to a z-score:
z = $\frac{x - \mu}{\sigma}$ = $\frac{24.9 - 25}{0.07}$ = -1.43
ii) Convert 25.1 to a z- score:
z = $\frac{25.1 - 25}{0.07}$ = 1.43
iii) $P(24.9 < x < 25.1) = P(-1.43 < z < 1.43)$
$P(-1.43 < z < 1.43) = P(z < 1.43) - P(z < -1.43) = 0.9236 - 0.0764 = 0.8472$
Therefore, $P(24.9 < x < 25.1) = 0.8472$
iv) 0.8472 of the total rods produced will be between 24.9 and 25.1 cm. We need 10,000 rods to be between 24.9 and 25.1 cm. If we let R be the number of rods produced:
0.8472 x R = 10,000
R = $\frac{10,000}{0.8472}$ = 11,803.588
Thus, to meet the order, the plant manager should produce 11,804 rods in total.
