Answer :
Answer:
482.2 - 100.8
≈ 381.4 cm^2
Step-by-step explanation:
For question 13:
We can begin by using the cosine rule to find the length of AC:
cos(A) = (b^2 + c^2 - a^2) / 2bc
cos(60) = (9^2 + AC^2 - 12^2) / (2*9*AC)
1/2 = (81 + AC^2 - 144) / (18AC)
9AC = AC^2 - 63
AC^2 - 9AC - 63 = 0
Solving this quadratic equation gives AC ≈ 11.2 cm (to 1 decimal place).
Next, we can use the sine rule to find the length of the CD:
sin(C) / CD = sin(A) / AC
sin(60) / CD = sin(80) / 11.2
CD = sin(60) * 11.2 / sin(80)
CD ≈ 7.5 cm (to 1 decimal place).
Therefore, (a) the length of AC ≈ is 11.2 cm, and (b) the length of CD ≈ is 7.5 cm.
For question 14:
The area of a regular pentagon can be found using the formula:
A = (5/4) * s^2 * cot(π/5)
where s is the side length.
In this case, the side length can be found using the radius of the circle:
s = 2 * r * sin(π/5)
s = 2 * 14 * sin(π/5)
s ≈ 14.4 cm (to 1 decimal place).
Substituting this into the formula for the area of a regular pentagon:
A = (5/4) * (14.4)^2 * cot(π/5)
A ≈ 482.2 cm^2 (to 4 significant figures).
The shaded region is formed by subtracting the area of the unshaded triangle from the area of the pentagon. The unshaded triangle is an isosceles triangle with a base length equal to a side of the pentagon and a height equal to the circle's radius. Therefore, its area is:
(1/2) * s * r
(1/2) * 14.4 * 14
≈ 100.8 cm^2 (to 1 decimal place).
So, the area of the shaded region is:
482.2 - 100.8
≈ 381.4 cm^2 (to 4 significant figures).
