Answer :
To determine the correct function that represents the total area [tex]\( A \)[/tex] of the window in terms of its width [tex]\( w \)[/tex], let's break down the problem step by step.
1. Rectangular Part:
- The width of the rectangle is [tex]\( w \)[/tex] meters.
- The height of the rectangle is [tex]\( 1 + 3w \)[/tex] meters.
- The area of the rectangle, [tex]\( A_{rectangle} \)[/tex], is given by:
[tex]\[ A_{rectangle} = \text{width} \times \text{height} = w \times (1 + 3w) = w(1 + 3w) \][/tex]
2. Semicircular Part:
- The semicircle is on top of the rectangle, so its diameter is equal to the width of the rectangle, which means:
[tex]\[ \text{Diameter} = w \implies \text{Radius} = \frac{w}{2} \][/tex]
- The area of a full circle with radius [tex]\( \frac{w}{2} \)[/tex] is:
[tex]\[ A_{\text{circle}} = \pi \left( \frac{w}{2} \right)^2 \][/tex]
- Since we only have a semicircle, we need half of this area:
[tex]\[ A_{\text{semicircle}} = \frac{1}{2} \pi \left( \frac{w}{2} \right)^2 = \frac{\pi}{2} \left( \frac{w^2}{4} \right) = \frac{\pi w^2}{8} \][/tex]
3. Total Area:
- The total area of the window, [tex]\( A \)[/tex], is the sum of the area of the rectangle and the area of the semicircle:
[tex]\[ A = A_{rectangle} + A_{semicircle} = w(1 + 3w) + \frac{\pi w^2}{8} \][/tex]
Now let's compare this expression with the given options:
A. [tex]\( A(w) = w(1 + 3w) + \pi\left(\frac{w}{2}\right)^2 \)[/tex]
B. [tex]\( A(w) = w(1 + 3w) + \frac{\pi\left(\frac{w}{2}\right)^2}{2} \)[/tex]
C. [tex]\( A(w) = w(1 + 3w) + \frac{\pi(w)^2}{2} \)[/tex]
D. [tex]\( A(w) = w(1 + 3w) + \pi(w)^2 \)[/tex]
- Option A: The term [tex]\( \pi\left(\frac{w}{2}\right)^2 \)[/tex] would correspond to the area of a full circle with diameter [tex]\( w \)[/tex], which is incorrect for a semicircle.
- Option B: This seems like a candidate, but let's simplify the term [tex]\( \frac{\pi\left(\frac{w}{2}\right)^2}{2} \)[/tex]:
[tex]\[ \text{Simplifying } \frac{\pi\left(\frac{w}{2}\right)^2}{2} = \frac{\pi \frac{w^2}{4}}{2} = \frac{\pi w^2}{8} \][/tex]
This gives us the correct semicircle area, so this option matches our derived formula.
- Option C: The term [tex]\( \frac{\pi (w)^2}{2} \)[/tex] is too large and is not the correct formula for the area of the semicircle.
- Option D: The term [tex]\( \pi (w)^2 \)[/tex] would be the area of a full circle with radius [tex]\( w \)[/tex], which again is incorrect.
So the correct answer is:
[tex]\[ \boxed{B} \][/tex]
Therefore, the function representing the total area, [tex]\( A \)[/tex], of the window in terms of the width [tex]\( w \)[/tex] is:
[tex]\[ A(w) = w(1 + 3w) + \frac{\pi\left(\frac{w}{2}\right)^2}{2} \][/tex]
1. Rectangular Part:
- The width of the rectangle is [tex]\( w \)[/tex] meters.
- The height of the rectangle is [tex]\( 1 + 3w \)[/tex] meters.
- The area of the rectangle, [tex]\( A_{rectangle} \)[/tex], is given by:
[tex]\[ A_{rectangle} = \text{width} \times \text{height} = w \times (1 + 3w) = w(1 + 3w) \][/tex]
2. Semicircular Part:
- The semicircle is on top of the rectangle, so its diameter is equal to the width of the rectangle, which means:
[tex]\[ \text{Diameter} = w \implies \text{Radius} = \frac{w}{2} \][/tex]
- The area of a full circle with radius [tex]\( \frac{w}{2} \)[/tex] is:
[tex]\[ A_{\text{circle}} = \pi \left( \frac{w}{2} \right)^2 \][/tex]
- Since we only have a semicircle, we need half of this area:
[tex]\[ A_{\text{semicircle}} = \frac{1}{2} \pi \left( \frac{w}{2} \right)^2 = \frac{\pi}{2} \left( \frac{w^2}{4} \right) = \frac{\pi w^2}{8} \][/tex]
3. Total Area:
- The total area of the window, [tex]\( A \)[/tex], is the sum of the area of the rectangle and the area of the semicircle:
[tex]\[ A = A_{rectangle} + A_{semicircle} = w(1 + 3w) + \frac{\pi w^2}{8} \][/tex]
Now let's compare this expression with the given options:
A. [tex]\( A(w) = w(1 + 3w) + \pi\left(\frac{w}{2}\right)^2 \)[/tex]
B. [tex]\( A(w) = w(1 + 3w) + \frac{\pi\left(\frac{w}{2}\right)^2}{2} \)[/tex]
C. [tex]\( A(w) = w(1 + 3w) + \frac{\pi(w)^2}{2} \)[/tex]
D. [tex]\( A(w) = w(1 + 3w) + \pi(w)^2 \)[/tex]
- Option A: The term [tex]\( \pi\left(\frac{w}{2}\right)^2 \)[/tex] would correspond to the area of a full circle with diameter [tex]\( w \)[/tex], which is incorrect for a semicircle.
- Option B: This seems like a candidate, but let's simplify the term [tex]\( \frac{\pi\left(\frac{w}{2}\right)^2}{2} \)[/tex]:
[tex]\[ \text{Simplifying } \frac{\pi\left(\frac{w}{2}\right)^2}{2} = \frac{\pi \frac{w^2}{4}}{2} = \frac{\pi w^2}{8} \][/tex]
This gives us the correct semicircle area, so this option matches our derived formula.
- Option C: The term [tex]\( \frac{\pi (w)^2}{2} \)[/tex] is too large and is not the correct formula for the area of the semicircle.
- Option D: The term [tex]\( \pi (w)^2 \)[/tex] would be the area of a full circle with radius [tex]\( w \)[/tex], which again is incorrect.
So the correct answer is:
[tex]\[ \boxed{B} \][/tex]
Therefore, the function representing the total area, [tex]\( A \)[/tex], of the window in terms of the width [tex]\( w \)[/tex] is:
[tex]\[ A(w) = w(1 + 3w) + \frac{\pi\left(\frac{w}{2}\right)^2}{2} \][/tex]
