Question

1. mude used a mug such as the one below to serve tea. the cup has a diameter of 7.7 cm and a height of 9.2 cm. mude added tea to half - half the capacity of the mug. what is the surface in contact with the tea?

Explanation:

Step1: Calculate the radius

The diameter $d = 7.7$ cm, so the radius $r=\frac{d}{2}=\frac{7.7}{2}=3.85$ cm.

Step2: Determine the height of the tea - level

The height of the mug $h = 9.2$ cm, and the tea is filled to half - half (presumably half) of the capacity, so the height of the tea $h_{tea}=\frac{9.2}{2}=4.6$ cm.

Step3: Calculate the lateral surface area in contact with tea

The formula for the lateral surface area of a cylinder is $A_{lateral}=2\pi rh$. For the tea in the mug, $A_{lateral}=2\pi\times3.85\times4.6$.
\[

$$\begin{align*} A_{lateral}&=2\times3.14\times3.85\times4.6\\ &=6.28\times3.85\times4.6\\ & = 24.148\times4.6\\ &=111.0808\text{ cm}^2 \end{align*}$$

\]

Step4: Calculate the area of the bottom surface in contact with tea

The formula for the area of the base of a cylinder is $A_{base}=\pi r^{2}$. So $A_{base}=3.14\times(3.85)^{2}=3.14\times14.8225 = 46.54265\text{ cm}^2$.

Step5: Calculate the total surface area in contact with tea

$A = A_{lateral}+A_{base}=111.0808 + 46.54265=157.62345\approx157.62\text{ cm}^2$

Answer:

$157.62\text{ cm}^2$