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Consider a cone of vertical height H (in metres) and base radius R (in metres) which is full with water. The cone, at time t=0, starts to leak such that it loses water at a rate of k m^3 per second. Give an expression for the rate of change of H.

L = (H 2 +R 2 ) 1/2 V = (1/3)πR 2 (H 2 +R 2 ) 1/2 dV/dt = -k dH/dt = dH/dV × dv/dt dV/dH = (1/3)πR 2 H(H 2 +R 2 ) 1/2 Thus, dH/dt = -3k/(πR 2 H(H 2 +R 2 ) 1/2 ) ​​​​​ ​​​​ ​​
CE
Answered by Callum E. • Maths tutor
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Integrate the following function: f(x) = ln(x)

∫ln(x)dx = xln(x) - ∫x/x dx = xln(x) - x + C
CE
Answered by Callum E. • Maths tutor
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Why is the differential of a constant zero?

Any constant (say k) can be rewritten as kx 0 since x 0 =1. When you differentiate this, the 0 which is the power of the x term gets dragged to the front and is multiplied with the rest of the expression (ac...
JC
Answered by Jawad C. • Maths tutor
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How do you find the maximum/minimum value of an equation?

When finding the maximum or minimum of an equation, make sure you rearrange an equation to make y the subject of the equation i.e. y=_________. You can then differentiate with respect to x and then set the d...
JC
Answered by Jawad C. • Maths tutor
7581 Views

Prove: (1-cos(2A))/sin(2A) = tan(A)

Firstly we must lay out our double angle formulas which are required for this question: cos(2A) = 1-2sin^2(A) = 2cos^2(A)-1 sin(2A) = 2sin(A)cos(A) Working from LHS: (1-cos(2A))/sin(2A) Focusing on the denom...
RP
Answered by Rishi P. • Maths tutor
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