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A rectangle is inscribed in a semicircle?
3.) A rectangle is inscribed in a semicircle (centered at the origin, above the x-axis) of radius 2. Let P=(x,y) be the point in quadrant 1 that is a vertex of the rectangle and is on the circle.
A.) Draw and accurately label a sketch of this situation.
B.) Express the area A of the rectangle as a function of x.
C.) Express the perimeter p of the rectangle as a function of x.
D.) use a graphing utility to graph A=A(x). For what value of x is A largest?
Round to the nearest tenth, if applicable. Provide a sketch of what you see in your grapher, including the window dimensions.
E.) Use a graphing utility to graph p=p(x). for what value of x is P largest?
Provide a sketch of what you see in your grapher, including the window dimensions.
Please help, I'm so lost on what to do for this.
1 Answer
- NickLv 67 years ago
A) draw and label rectangle inscribed in semi circle with r=2 centered at origin.
B) notice, distance from origin along x axis to RH lower corner of rectangle = x this is 1/2 the base so base = 2x.
equation of semicircle:
x^2+y^2=r^2
x^2+y^2=4
y = sqrt(4 - x^2)
also notice that since height of rectangle terminates on semi circle then height is y or sqrt(4 - x^2) hence:
Area = base * height
Area = 2x(sqrt(4 - x^2)
C) p = 2(2x) + 2(sqrt(4 - x^2)
p = 4x + 2(sqrt(4 - x^2)
