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We use the quotient rule here which states that if y = f(x)/g(x) then dy/dx = (f'(x)g(x) - g'(x)f(x)) / (g(x)^2). Here f(x) = 4x and g(x) = x^2 + 5, so we have f'(x) = 4 , g'(x) = 2x. This gives us dy/dx ...

We know that an equation has equal roots if the sqrt(b^2-4ac) term in the quadratic equation is equal to zero. Therefore using this information we can form an expression for k to be (-k)^2-4(2k-3)(k-1)=0....

As V=(h⁶+16)^1/2 -4, the chain rule can be used to calculate dV/dh.

dV/dh=3h⁵(h⁶+16)^-1/2

h=2 can be su...

51/100 * 400 = 51/100 * 400/1 = 51/1 * 4/1

= 204 

Solve this simultaneously.

Where a= apples and p= pears,

a+p=50 and 2a+3p=130

Multiply a+p=50 by 2 on both sides to get:

2a+2p=100

Then subtract the two equations from e...