Differentiation, small chance and line graph questions?

Question

1. sand is poured onto horizontal ground. the sand forms a right circular cone. the volume of the cone increases in such a way that the height is always half the radius of the base. Find an estimate for the percentage change in in the volume of the cone when the radius increases by one percent.

2. A curve has equation y=3+ (1/x+3) where x is not equal to -3. 
a) find an equation of the asymptote to the curve which is parallel to 
i) the x axis 
ii) the y axis 
b) calculate the coordinates of the points where the curve crosses the coordinate axes

gile · Accepted Answer

1.
V = π r² h/3
Since h = r/2,
V = π r³/6
dV = 3π r²dr/6 = π r²dr/2
dV/V = π r²dr/2V
. . . . = π r²dr/(2π r³/6)
. . . . = 6 π r²dr/2 π r³
. . . . = 3 dr/t
As dr/r = 1%, dV/V = 3×1% = 3%

2.
y = 3 + 1/(x + 3)
a) As x→ ±∞, y→3, hence y = 3 is the horizontal asymptote to the curve
b) As x→ −3, y→∞, hence x = −3 is the vertical asymptote to the curve
c) 
* As x = 0, y = 4, hence the curve crosses the y-axis at (0,4)
* For y to be 0, x must satisfy 3 + 1/(x + 3) = 0, from there 3(x + 3) + 1 = 0
Solving for x: 3x + 9 + 1 = 0 ⇒ x = -10/3. 
Hence the curve crosses the x-axis at (-10/3,0)

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Differentiation, small chance and line graph questions?

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