Ted:
4,000 @ 4.5% interest compounded annually

Alan:
1,000 @ 6% interest compounded annually

Rule of 72: divide 72 by the compound annual interest rate

Ted: 72/4.5% = 16 years
Alan: 72/6% = 12 years

<span>Alan will win. It will take 12 years.</span>

8 0

3 years ago

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Find an equation for the plane that is tangent to the surface z equals ln (x plus y )at the point Upper P (1 comma 0 comma 0 ).

alexira [117]

Let $f(x,y,z)=z-\ln(x+y)$ . The gradient of $f$ at the point (1, 0, 0) is the normal vector to the surface, which is also orthogonal to the tangent plane at this point.

So the tangent plane has equation

$\nabla f(1,0,0)\cdot(x-1,y,z)=0$

Compute the gradient:

$\nabla f(x,y,z)=\left(\dfrac{\partial f}{\partial x},\dfrac{\partial f}{\partial y},\dfrac{\partial f}{\partial z}\right)=\left(-\dfrac1{x+y},-\dfrac1{x+y},1\right)$

Evaluate the gradient at the given point:

$\nabla f(1,0,0)=(-1,-1,1)$

Then the equation of the tangent plane is

$(-1,-1,1)\cdot(x-1,y,z)=0\implies-(x-1)-y+z=0\implies\boxed{z=x+y-1}$

7 0

3 years ago