Find the points on the surface y^2 = 9 + xz that are closest to the origin.

This question aims to learn the basic methodology for optimizing a mathematical function (maximizing or minimizing). Critical points are the points where the value of a function is either maximum or minimum. To calculate the critical point(s), we equate the first derivative’s value to 0 and solve for the independent variable. We can use the […]

A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is a maximum?

This question aims to find the total area enclosed by a wire when it is cut down into two pieces. This question uses the concept of the area of a rectangle and an equilateral triangle. The area of a triangle is mathematically equal to: [Area space of space triangle space = space frac{Base space times space […]

For the equation, write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. Keeping the restrictions in mind, solve the equation.

(dfrac{4}{x+5}+dfrac{2}{x-5}=dfrac{32}{x^2-25}) This question aims to find the solution to the given equation by taking into consideration the restrictions on the given function. The fraction of two polynomials is said to be a rational expression. Such expression can be expressed as $dfrac{a}{b}$ in which $a$ and $b$ both are polynomials. The product, sum, division, and subtraction […]

Find the parametric equation of the line through a parallel to b.

(a=begin{bmatrix}3\-4end{bmatrix}, b=begin{bmatrix}-7\8end{bmatrix}) This question aims to find the parametric equation of the line through two given vectors. A parametric equation is an equation that incorporates a parameter that is an independent variable. In this equation, the dependent variables are the continuous functions of the parameter.  Two or more parameters can also be used if needed. […]