Alan

"The sum of 3 real numbers is known to be zero. If the sum of their cubes is -1, what is their product equal to?"

"The sum of lengths of all edges of a rectangular box is  140 cm and the distance from one corner
of the box to the farthest corner is 21 cm.  What is the total surface area of the box?"

In what sense do you want to "solve" this?  You can plot f as a function of x:

I suspect the second part should say the ratio of books to pens is now 2:3, in which case:

With p = number of pens and b = number of books we have:

The following should help:

Note that f(x) = 1 has exactly 6 solutions (draw a line through y = 1 and count how many times it cuts through the curve).

Also, f(x) = 2 has exactly 6 solutions (draw a line through y = 2 and count how many times it cuts through the curve).

Also, f(x) = ...

Continue in this way until the line through the integer value of y does not cut through the curve exactly 6 times.

Because you said there is one unit between lines, I've assumed the fine grid is numbered 1, 2 etc.  However, your labels give the coarse grid as  having labels 1, 2 etc, in which case, only the value f(x) = 1, is relevant.

We have \(z^{49}+z^{50}+z^{51}+z^{52}=z^{49}(1+z+z^2+z^3)\)

The term in parentheses factors as \(1+z+z^2+z^3=(z+1)(z^2+1)\)

So, if \(z^2+1=0\) then \(z^{49}+z^{50}+z^{51}+z^{52}=0\)

"If x and y are positive integers for which 3x + 2y + xy = 115 + 7x, then what is x + y?"

I suspect the first term of h(t) should be -4.9t^2  not -4.9t.

Then by setting h(t) = 6 you have a quadratic equation which will give you two solutions for t; the difference between them being the length of time required.

The following also work:

  \(a=\frac{2}{\sqrt3}, b=\frac{1}{\sqrt3}\\a=-\frac{2}{\sqrt3}, b = -\frac{1}{\sqrt3}\)