Give a composite shape of quarter of circle and rectangle with a total area of #570 " square feet"# and and the diagonal angle of the rectangle equal to #18.43^0#, calculate the radius?

Question

Theodore Abad · Accepted Answer

#"Radius"~~22.92 "ft".# Let #r# be the radius. Then, using the Figure given, #IH=rcos theta, IA=rsin theta, "where, "theta=18.43^@#. #:." Area of the Rectangle "IHBA=IH*IA=r^2sin thetacos theta#. #=r^2/2*sin 2theta=r^2/2*(2tan theta)/(1+tan^2 theta)# #=(r^2tan theta)/(1+tan^2 theta)# For #theta=18.43^@, tan theta=1/3# #"Hence, Area of the Rectangle"=(r^2/3)/(1+1/9)=3r^2/10# #"Also, Area of the Quarter of the Circle"=1/4*pir^2# #:. "Total Area of the Composite Shape"=3r^2/10+pir^2/4=r^2/20(6+5pi)#, which is, #570" sq.ft."# #:. r^2/20(6+5pi)=570# #:. r^2=(570*20)/(6+5pi)=11400/(6+15.7)=11400/21.7~~525.3456# #:. r~~22.92 "ft."#

Evelyn Arboleda · Answer

undefined To find the radius of the quarter circle, we first need to find the dimensions of the rectangle. Let's denote the length of the rectangle as $ l $ and the width as $ w $.

The area of the rectangle is given by $ A_{\text{rectangle}} = l \times w $.

Given that the total area of the composite shape is 570 square feet, and one quarter of this area is occupied by the quarter circle, the area of the rectangle is $ \frac{3}{4} \times 570 $.

So, we have:

$$ A_{\text{rectangle}} = \frac{3}{4} \times 570 $$

Now, let's find the length and width of the rectangle using the fact that the diagonal angle of the rectangle is $ 18.43^\circ $.

The diagonal of the rectangle can be expressed as:

$$ \text{Diagonal} = \sqrt{l^2 + w^2} $$

Given that the diagonal angle of the rectangle is $ 18.43^\circ $, we can express this as:

$$ \tan(18.43^\circ) = \frac{w}{l} $$

From this, we can solve for $ w $ in terms of $ l $:

$$ w = l \times \tan(18.43^\circ) $$

Substitute this into the equation for the area of the rectangle:

$$ \frac{3}{4} \times 570 = l \times (l \times \tan(18.43^\circ)) $$

Solve this equation for $ l $, then use the obtained value of $ l $ to find $ w $. Once you have both the length and width of the rectangle, you can find the radius of the quarter circle using the fact that its area is one quarter of the total area and $ A_{\text{quarter circle}} = \frac{1}{4} \pi r^2 $, where $ r $ is the radius of the quarter circle.