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Chapter 11: Problem 16

Solve each formula for the indicated letter. Assume that all variables represent nonnegative numbers. \(A=6 s^{2},\) for \(s\) (Surface area of a cube with sides of length \(s\) )

Short Answer

Expert verified
s = \sqrt{\frac{A}{6}}.

Step by step solution

01

- Write the given formula

The formula given is for the surface area of a cube: \( A = 6s^{2} \) We need to solve for \( s \).
02

- Isolate the term with the variable

To solve for \( s \), we first need to isolate \( s^{2} \). Do this by dividing both sides of the equation by 6: \( \frac{A}{6} = s^{2} \).
03

- Solve for the variable

To solve for \( s \), take the square root of both sides: \( s = \sqrt{\frac{A}{6}} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

surface area
Surface area is the total area covered by the surface of a three-dimensional object. Imagine wrapping a present; the paper you use represents the surface area. Knowing how to calculate surface area helps in understanding the size and space an object occupies, which is crucial in various fields like architecture, packaging, and more. For different shapes, the formulas to calculate surface area vary. For instance, the surface area of a cube is calculated differently from that of a sphere or a cylinder. Let鈥檚 dive deeper into the cube's surface area in the next section.
isolating variables
Isolating variables is a fundamental step in algebra that involves manipulating an equation to solve for one specific variable. This can be particularly useful when dealing with formulas in different scenarios, such as finding the side length of a cube from its surface area. Here鈥檚 a simplified process:
  • Identify the variable you need to isolate.
  • Use algebraic operations (addition, subtraction, multiplication, division) to get the variable by itself on one side of the equation.
  • If dealing with squares or higher powers, consider operations like taking roots or inverse operations.
In our example, we isolate the variable s by first dividing the surface area formula by 6 and then taking the square root. These steps make the process systematic and easier to follow.
cube surface area formula
The surface area formula for a cube is given by: \(A = 6s^{2}\). This formula comes from the fact that a cube has six identical square faces. Here鈥檚 a breakdown:
  • Each face of the cube has an area of \(s^{2}\), where s is the side length.
  • Since there are six such faces, multiply the area of one face by 6.
  • The product, 6s2, gives the total surface area of the cube.
To solve for s using this formula, you鈥檒l isolate s by dividing both sides by 6 (which gives you \( \frac{A}{6} = s^{2} \)), and then taking the square root of both sides to find \( s = \sqrt{\frac{A}{6}} \). This simple yet effective formula is quite handy in quickly determining the side length of a cube when its surface area is known.

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Most popular questions from this chapter

Show that whenever there is just one solution of \(a x^{2}+b x+c=0,\) that solution is of the form \(-b /(2 a)\).

For each equation under the given condition, (a) find \(k\) and (b) find the other solution. Find an equation with integer coefficients for which \(1-\sqrt{5}\) and \(3+2 i\) are two of the solutions.

Solve by completing the square. Show your work. $$ t^{2}-10 t=-23 $$

Let \(f(t)=(t+6)^{2} .\) Find \(t\) such that \(f(t)=15\).

Solve. $$ 9 x^{2}-16=0 $$
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