Question

tim is an elementary school art teacher. his students are sculpting a replica of a shark out of clay. tim has given them one block of clay to make 20 conical shark teeth for the sculpture. the block contains 81 cm³ of clay. if each tooth is solid and has a 2.5 cm base diameter, what is the maximum height each tooth can be? assume that the students use all the clay. a. 1.25 cm b. 2.47 cm c. 3.57 cm d. 2.27 cm e. 4.50 cm

Explanation:

Step1: Find the volume formula for a cone

The volume formula for a cone is $V = \frac{1}{3}\pi r^{2}h$, where $r$ is the radius and $h$ is the height. Given the base - diameter $d = 2.5$ cm, the radius $r=\frac{d}{2}=\frac{2.5}{2}=1.25$ cm.

Step2: Calculate the volume of one tooth

Let the height of each tooth be $h$. The volume of one conical tooth is $V=\frac{1}{3}\pi(1.25)^{2}h$.

Step3: Determine the total volume of clay available

The total volume of clay is $81$ $cm^{3}$ and it is used to make 20 teeth. So the volume of clay available for each tooth is $V_{each}=\frac{81}{20}=4.05$ $cm^{3}$.

Step4: Solve for the height $h$

Set $\frac{1}{3}\pi(1.25)^{2}h = 4.05$. First, calculate $(1.25)^{2}=1.5625$. Then $\frac{1}{3}\times3.14\times1.5625h = 4.05$. $\frac{1}{3}\times3.14\times1.5625h=1.63541667h$. So, $h=\frac{4.05}{1.63541667}\approx2.47$ cm.

Answer:

B. 2.47 cm