Question

Exercise 102

Suppose a ball is dropped from a height of 200m. If time after the ball was released is in seconds, and the height of the ball above the ground at time is given by

 , find the average velocity from to 

Show that the slope of the line joining the points A(1,1) and B is 

A line passes through the points (0, 5) and (9, −1). Find the equation of the line which is perpendicular to the line and passes through its midpoint.

Find the equation of the line that is parallel to the line y = −2x + 6 and passing through the point A(1, 10).

A square has vertices O(0, 0), A(a, 0), B(a, a) and C(0, a).

i Find the midpoint of the diagonals OB and CA.

ii Find the length of a diagonal of the square and the radius of the circle in which OABC is inscribed.

Iii Find the equation of the circle inscribing the square

Answer 1

1. $h(t)=200-10t^2$

$$h(2)=200-10(2)^2=160$$

$$h(3)=200-10(3)^2=110$$

The average velocity from 2s to 3s is

$$v_{ave}=\dfrac{110m-160m}{3s-2s}=-50m/s$$

2.

$$A(1, 1), B(2, 3+h)$$

$$slope=\dfrac{y_B-y_A}{x_B-x_A}=\dfrac{3+h-1}{2-1}=2+h$$

3. Midpoint $(9/2, 2)$

$$slope_1=\dfrac{-1-5}{9-0}=-\dfrac{2}{3}$$

If line is perpendicular then

$$slope_2=-1/(-\dfrac{2}{3})=\dfrac{3}{2}$$

$$y=\dfrac{3}{2}x+b$$

$$2=\dfrac{3}{2}(\dfrac{9}{2})+b=>b=-\dfrac{19}{4}$$

The equation of the line is

$$y=\dfrac{3}{2}x-\dfrac{19}{4}$$

4.

$$y=-2x+b$$

$$10=-2(1)+b=>b=12$$

The equation of the line is

$$y=-2x+10$$

5.

i.

$$M(a/2, a/2)$$

ii.

$$AC=BD=a\sqrt{2}, >0$$

$$r=a/2, a>0$$

iii.

$$R=OB/2=\dfrac{\sqrt{2}a}{a}, a>0$$