The diagram shows part of the curve y = 8/(square root of 3x+4). The curve intersects the y-axis at A(0,4). The normal to the curve at A intersects the line x=4 at the point B.

Question

i. Find the coordinates of B.

correct answer: (4, 20/3)

ii. Show, all necessary working, that the areas of the regions P and Q are equal.

correct answer: P and Q are both 32/3

Answer 1

The derivative (y') is -12/(√(3x+4))³.

So the slope of the tangent line when x = 0 is -3/2. That means the normal line has a slope of 2/3.

So the equation of the normal line is y = 2x/3 + 4. That line intersects the line x = 4, when in fact x = 4.

When x = 4, the y-coordinate of the normal line is 2(4)/3 + 4 = 8/3 + 12/3 = 20/3.

The following graph shows regions P and Q having an area of 32/3.

Check rows 12 and 13 to see how the integrals for the areas of those regions are set up.

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