Solve the problem.
-A manufacturer contracts to mint coins for the federal government. How much variation dr in the radius of the coins can be tolerated if the coins are to weigh within $1 / 500$ of their ideal weight? Assume that the thickness does not vary.
A) 0.20%
B) 0.10%
C) 0.0010%
D) 0.0020%

Question

Quizplus · Accepted Answer

The weight of a coin, assuming constant thickness and density, is directly proportional to its volume, which for a cylinder (a simplified coin shape) is proportional to the square of its radius. A $1/500$ variation in weight implies a $\sqrt{1/500}$ variation in radius, because the square root function is the inverse of squaring in the proportional relationship. Calculating $\sqrt{1/500}$ gives approximately 0.0447, which is about 4.47%. However, since the question likely involves a misunderstanding in the calculation (possibly looking for a much smaller variation due to the options provided), the correct approach to match the given options would involve considering the direct proportionality to the square of the radius and the small percentage change approximation formula. For small changes, if $y = kx^2$, then $\frac{\Delta y}{y} \approx 2\frac{\Delta x}{x}$, where $k$ is a constant, $x$ is the original value, $\Delta x$ is the change in $x$, $y$ is the original value after applying the function, and $\Delta y$ is the change in $y$. Given $\frac{\Delta y}{y} = \frac{1}{500}$, solving for $\frac{\Delta x}{x}$ gives $\frac{1}{1000}$, or 0.1%, indicating a mistake in the initial calculation and interpretation. The correct interpretation involves understanding the direct relationship between the square of the radius and the volume (and thus weight) and applying the small percentage change approximation, leading to the conclusion that a 0.1% variation in radius can be tolerated to maintain the weight within $1/500$ of its ideal value.

Solve the Problem 1/5001 / 5001/500 Of Their Ideal Weight? Assume That the Thickness Does Not

Solve the Problem $1 / 500$ Of Their Ideal Weight? Assume That the Thickness Does Not