Question - Calculating Area of a Rose Garden
Solution:
To find the area of this rose garden, you need to calculate the area of the rectangle and add the areas of the two semi-circles on either end of the rectangle.First, calculate the area of the rectangle. The area $$ A_{rectangle} $$ is given by the formula:\[ A_{rectangle} = \text{length} \times \text{width} \]For this garden:\[ A_{rectangle} = 21 \text{ ft} \times 14 \text{ ft} = 294 \text{ ft}^2 \]Now calculate the area of a full circle and then take half of it to get the area of a semi-circle. The diameter of the circles is the width of the rectangle, which is 14 feet. The radius $$ r $$ is therefore 7 feet (since radius is half the diameter). The area $$ A_{circle} $$ of a full circle is given by the formula:\[ A_{circle} = \pi r^2 \]But since we have two semi-circles, their combined area will equal the area of one full circle. So we will use the formula for a full circle and not divide it by 2 in the end. Let's calculate this area using $$ \pi \approx 3.14 $$:\[ A_{circle} = 3.14 \times 7^2 = 3.14 \times 49 = 153.86 \text{ ft}^2 \]So, to find the total area of the garden $$ A_{garden} $$, add the area of the rectangle and the area of the circles:\[ A_{garden} = A_{rectangle} + A_{circle} \]\[ A_{garden} = 294 \text{ ft}^2 + 153.86 \text{ ft}^2 = 447.86 \text{ ft}^2 \]Now, round to the nearest whole number if necessary and do not round your answer:\[ A_{garden} \approx 448 \text{ ft}^2 \]Therefore, the area of the garden is approximately 448 square feet.
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