Question

1. the density of the american white oak tree is 752 kilograms per cubic meter. if the trunk of an american white oak tree has a circumference of 4.5 meters and the height of the trunk is 8 meters, what is the approximate number of kilograms of the trunk?
2. a garden bed, pictured below, is a square prism with a rectangular prism taken out. the inside length of the square prism is 6 feet. the rectangular prism taken out has a width of 2 feet and a length of 4 feet. the diagram below shows the top view of the garden bed with its inside measurements.

the garden bed is filled with topsoil to a uniform height of 1.25 feet.
determine and state the volume of the topsoil, in cubic feet.
each bag of topsoil sells for $3.68 and contains 2 cubic feet of topsoil.
determine and state the total cost of the bags of topsoil that must be purchased to fill the garden.

Explanation:

Step1: Find radius of tree trunk

Use formula $C = 2\pi r$, so $r=\frac{C}{2\pi}$. Given $C = 4.5$ meters, then $r=\frac{4.5}{2\pi}$ meters.

Step2: Calculate volume of tree trunk

Volume of cylinder $V=\pi r^{2}h$. Substitute $r = \frac{4.5}{2\pi}$ and $h = 8$ meters. $V=\pi(\frac{4.5}{2\pi})^{2}\times8=\pi\times\frac{4.5^{2}}{4\pi^{2}}\times8=\frac{4.5^{2}\times8}{4\pi}=\frac{20.25\times8}{4\pi}=\frac{162}{4\pi}\approx12.89$ cubic - meters.

Step3: Calculate mass of tree trunk

Use formula $m=
ho V$. Given $
ho = 752$ kg/m³ and $V\approx12.89$ m³. Then $m = 752\times12.89=752\times\frac{162}{4\pi}\approx9693.28$ kilograms.

Step4: Calculate volume of top - soil in garden bed

Volume of outer square prism $V_{1}=6\times6\times1.25 = 45$ cubic feet. Volume of inner rectangular prism $V_{2}=4\times2\times1.25 = 10$ cubic feet. Volume of top - soil $V = V_{1}-V_{2}=45 - 10=35$ cubic feet.

Step5: Calculate number of top - soil bags

Number of bags $n=\frac{V}{2}=\frac{35}{2}=17.5$. Since we can't buy half - a - bag, we need to buy 18 bags.

Step6: Calculate total cost of top - soil bags

Cost per bag is $\$3.68$. Total cost $C=3.68\times18=\$66.24$.

Answer:

The mass of the tree trunk is approximately $9693.28$ kilograms. The volume of the top - soil is 35 cubic feet. The total cost of the top - soil bags is $\$66.24$.