Question

Suppose you have a coffee mug with a circular cross section and vertical sides (uniform radius). What is its inside radius if it holds $375g$of coffee when filled to a depth of$750cm$? Assume coffee has the same density as water.

Short answer

The radius of the cup is obtained as: $3.988cm$.

Step-by-step solution

Conceptual Introduction

Fluid statics, often known as hydrostatics, is a branch of fluid mechanics that investigates the state of balance of a floating and submerged body, as well as the pressure in a fluid, or imposed by a fluid, on an immersed body.

Given Data

The density of a material is defined as the ratio of its mass to its volume.

The density of water is$\rho ={\text{1 g/cm}}^{\text{3}}$.

When we will have the coffee mug with some amount of water. So, it will be having certain volume.

The cup is in cylindrical form so, the volume of the cup will be$\pi r^{2}h$.

Here radius is not given.

The height of the cup is given$h=\text{7.5 cm}$.

The amount of the coffee present in the cup $m=\text{375 g}$.

Radius of the cup

As we know that the density is the rate of mass per volume let’s put the given data in the formula given below.

$$\begin{gathered} \rho =\frac{m}{V} \\ 1{\text{g/cm}}^3=\frac{375 g}{\pi r^{2}h} \end{gathered}$$

Now, calculating the value of r,

$$\begin{gathered} r^2=\frac{375 g}{\pi h} \\ {r}^2=\frac{375 g}{\frac{22}{7}\times 7.5} \\ {r}^2=15.909{\text{cm}}^2 \\ r=3.988 \text{cm} \end{gathered}$$

Therefore, the radius of the cup is: $3.988cm$.