# integration and coordinate geometry

• Apr 1st 2006, 09:56 PM
devilicious
integration and coordinate geometry
sorry i was going through some past papers and i need help on some questions again.

1)
given that dy/dx = ((3-square root x)^2)/square root x, x>0, and that y=2/3 at x=1
Find y in terms of x

2)
the line l2 passes through the origin O and has gradient -2. the lines l2 and y=1/3x-7 intersect at the point P.
a) calculate the coordinates of P

given that y=1/3x-7 crosses the y-axis at the point C,
b) calculate the exact area of triangeOCP.

Thanks!
• Apr 1st 2006, 11:30 PM
earboth
Quote:

Originally Posted by devilicious
sorry i was going through some past papers and i need help on some questions again.
1)
given that dy/dx = ((3-square root x)^2)/square root x, x>0, and that y=2/3 at x=1
Find y in terms of x

2)
the line l2 passes through the origin O and has gradient -2. the lines l2 and y=1/3x-7 intersect at the point P.
a) calculate the coordinates of P

given that y=1/3x-7 crosses the y-axis at the point C,
b) calculate the exact area of triangeOCP.

Thanks!

Hello,

to 1.: Integrate dy/dx and you'll get:
$\displaystyle \int{\frac{(3-\sqrt{x})^2}{\sqrt{3}}}dx=\frac{2}{3} \cdot (\sqrt{x}-3)^3+c$
Plug in the values you know and solve for c. You'll get c=6. So the equation of your function reads:
$\displaystyle f(x)=\frac{2}{3} \cdot (\sqrt{x}-3)^3+6$

to 2.: Use the slope-point-formula and you'll get : l2: y=-2x. Plug in this term for y into the 2nd equation: -2x=1/3x-7. Solve for x. x = 3. Sub in this value in one of the two equations and you'll get y = -6. So the coordinates of P are (3, -6).

to 3.: C has the coordinates (0, -7). Therefore the base of the triangle lies on the y-axis and has a length of 7. The height of the triangle is the x-value of point P. Now use the formula for the area of a triangle:
$\displaystyle A=\frac{1}{2} \cdot b \cdot h\ \Rightarrow \ A=\frac{1}{2} \cdot 7 \cdot 3 = \frac{21}{2}$

Greetings

EB
• Apr 2nd 2006, 08:28 PM
earboth
Quote:

Originally Posted by devilicious
sorry i was going through some past papers and i need help on some questions again.
...
2)
the line l2 passes through the origin O and has gradient -2. the lines l2 and y=1/3x-7 intersect at the point P.
a) calculate the coordinates of P
given that y=1/3x-7 crosses the y-axis at the point C,
b) calculate the exact area of triangeOCP.

Thanks!

Hello,

I've attached a diagram to illustrate my solution on problem 2.a) and 2.b)

Greetings

EB