In year 13, the scientist will put tree wrap around tree 1 to protect it from the winter snow. The height of the tree wrap needs to be 45 inches. The wrap is priced by the square foot. To the nearest square foot, how many square feet of wrap does she need?
Tree 1
\begin{tabular}{|c|c|}
\hline Year & \begin{tabular}{c}
Trunk \\
Diameter \\
(inches)
\end{tabular} \\
\hline 1 & 18.6 \\
\hline 3 & 19.2 \\
\hline 5 & 19.8 \\
\hline 7 & 20.4 \\
\hline 9 & 21.0 \\
\hline 11 & 21.6 \\
\hline 13 & 22.2 \\
\hline
\end{tabular}
Tree 2
\begin{tabular}{|c|c|}
\hline Year & \begin{tabular}{c}
Trunk \\
Diameter \\
(inches)
\end{tabular} \\
\hline 1 & 11.4 \\
\hline 3 & 12.0 \\
\hline 5 & 12.6 \\
\hline 7 & 13.2 \\
\hline 9 & 13.8 \\
\hline 11 & 14.4 \\
\hline 13 & 15.0 \\
\hline
\end{tabular}
Answer (2)
To protect Tree 1 in year 13, approximately 22 square feet of tree wrap is needed. This calculation is based on the trunk's diameter and the required wrap height. The area calculation involves finding the circumference first, converting units, and multiplying by the height of the wrap.
;
Calculate the circumference of the tree trunk using the formula C = π d , where d = 22.2 inches.
Convert the height of the tree wrap from inches to feet: h = 12 45 = 3.75 feet.
Calculate the area of the tree wrap: A = C × h = ( π × 22.2/12 ) × 3.75 × 12 ≈ 21.79 square feet.
Round the area to the nearest square foot: 22 square feet.
Explanation
Problem Analysis The problem asks us to find the square footage of tree wrap needed to protect tree 1 during year 13. We are given that the height of the wrap is 45 inches and the price is per square foot. The table provides the trunk diameter of tree 1 in year 13, which is 22.2 inches.
Calculations First, we need to calculate the circumference of the tree trunk. The formula for the circumference of a circle is C = π d , where d is the diameter. In this case, the diameter is 22.2 inches. So, the circumference is: C = π × 22.2 ≈ 69.73 inches Next, we need to convert the height of the tree wrap from inches to feet. Since there are 12 inches in a foot, we divide the height by 12: h = 12 inches/foot 45 inches = 3.75 feet Now, we can calculate the area of the tree wrap needed. The area is the circumference of the trunk multiplied by the height of the wrap in feet: A = C × h = 69.73 inches × 3.75 feet Since the circumference is in inches, we need to convert it to feet as well: C = 12 inches/foot 69.73 inches ≈ 5.81 feet Now we can calculate the area: A = 5.81 feet × 3.75 feet ≈ 21.79 square feet Finally, we round the area to the nearest square foot: A ≈ 22 square feet
Final Answer The area of the tree wrap needed is approximately 22 square feet.
Examples
Imagine you're decorating a cylindrical birthday cake and want to wrap a ribbon around it. Knowing the cake's diameter and the desired ribbon height, you can calculate the ribbon length (circumference) and the total ribbon area needed, similar to calculating the tree wrap area. This helps ensure you have enough ribbon to decorate the cake beautifully. For example, if the cake has a diameter of 10 inches and you want the ribbon to be 3 inches high, you would calculate the circumference as C = π × 10 ≈ 31.4 inches. The area of the ribbon needed would be A = 31.4 × 3 = 94.2 square inches.
