1. Find a maximum area of the rectangle that can be inscribed in the right trian

Question

1. Find a maximum area of the rectangle that can be inscribed in the right triangle with legs of length 3 and 4.

2. A particle moves in a straight line and has acceleration given by a(t)=4t-1.
Find a position of the particle, s(t), given that v(0)=-2 and s(0)=1.

3. Find the slope of the line tangent to the curve of the function G(x)= integral of (t+1)^1/2 at t=2 with the upper limit x^3 and lower 1.

4. Set up but do not evaluate an integral that represents the area of the region bounded by y=x+3 and y=23-x^2

5. Find the volume of the solid obtained by rotating the region in the first quadrant bounded by the curve y=x^2 and y=3 about the y-axis

6. Set up but do not evaluate an integral that represents the volume of the solid of revolution obtained by rotating the region bounded by the curves y=3x-x^2 and y=0 about the line x=-1

Explanation / Answer

1)Call the horizontal rectangle side y and the height of the rectangle x

The sides of the rectangle are parallel to the 90 degree legs. The top of the rectangle forms a similar triangle within (3,4,5).

So 4/3 = (4-x) / y
4y = 3(4-x) = 12 - 3x
y = 12/4 - 3x/4
y = 3 - 3x/4

Are of rectangle R = xy = x(3 - 3x/4) = 3x - 3x

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