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A conical glass holds 131 cubic centimeters of water. It has a height of 5 centimeters. The radius of the top of the glass is sqrt((m)/(5pi)) centimeters, where m is a constant. What is the value of m?

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A conical glass holds $131$ cubic centimeters of water. It has a height of $5$ centimeters. The radius of the top of the glass is $\sqrt{\left(\frac{m}{5\pi}\right)}$ centimeters, where $m$ is a constant. What is the value of $m$ ? $\newline$ $\square$

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Solve quadratic equations: word problems

Full solution

Q. A conical glass holds

131

cubic centimeters of water. It has a height of

5

centimeters. The radius of the top of the glass is

\sqrt{\left(\frac{m}{5\pi}\right)}

centimeters, where

m

is a constant. What is the value of

m

?

\newline

\square

Volume Formula Substitution: The volume of a cone is given by the formula $V = \frac{1}{3}\pi r^2h$ , where $V$ is the volume, $r$ is the radius, and $h$ is the height. We are given that the volume $V$ is $131$ cubic centimeters and the height $h$ is $5$ centimeters. We need to find the value of $m$ in the expression for the radius $r = \sqrt{\frac{m}{5\pi}}$ .
Isolating $r^2$ : First, let's substitute the given values into the volume formula and solve for $r^2$ . $131 = (\frac{1}{3})\pi r^2(5)$
Calculating $r^2$ : To isolate $r^2$ , we multiply both sides by $3$ and divide by $5\pi$ . $\newline$ $r^2 = \frac{3 \times 131}{5\pi}$
Setting Expressions Equal: Now, we calculate the value of $r^2$ . $\newline$ $r^2 = \frac{393}{5\pi}$
Final Solution: We are given that $r = \sqrt{\frac{m}{5\pi}}$ , so $r^2 = \frac{m}{5\pi}$ . We can set the two expressions for $r^2$ equal to each other to solve for $m$ . $\frac{m}{5\pi} = \frac{393}{5\pi}$
Final Solution: We are given that $r = \sqrt{\frac{m}{5\pi}}$ , so $r^2 = \frac{m}{5\pi}$ . We can set the two expressions for $r^2$ equal to each other to solve for $m$ . $\frac{m}{5\pi} = \frac{393}{5\pi}$ Since the $(5\pi)$ terms are on both sides of the equation, they cancel out, leaving us with $m = 393$ . $m = 393$

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