Question

If \(V=A h\) find the value of \(V\) when \(A=3\) and \(h=26\).

Short answer

Answer: The volume V is 78.

Step-by-step solution

Write down the formula

The formula to find the volume V is: $$V = A \cdot h$$

Substitute the given values into the formula

We are given that A = 3 and h = 26, so we can substitute these values into the formula: $$V = 3 \cdot 26$$

Calculate V

Now, just multiply the two numbers to find the value of V: $$ V = 3 \cdot 26 = 78$$ So, when A = 3 and h = 26, the value of V is 78.

Key concepts

Volume formula

Understanding the volume formula is crucial when trying to solve problems involving three-dimensional space. Generally, volume is the amount of space that an object occupies, and it is measured in cubic units, such as cubic meters or cubic centimeters. Different shapes have different formulas to calculate their volume. For example, the volume of a cube is found by raising the length of a side to the third power, while the volume of a cylinder is calculated by multiplying the area of its base (a circle) by its height.

In the given exercise, we deal with a generic volume formula, represented as

\[ V = A \cdot h \]
where V stands for volume, A is the area of the base, and h represents the height of the object. It’s essential to understand that this formula is versatile and can be used for various shapes as long as you know the area of the base and the height of the object. For the solid in the exercise, once we have the values for A and h, we can directly substitute them into the formula to find the volume.

Substitution method

The substitution method is a fundamental technique used in algebra to solve equations and simplify expressions. It involves replacing variables with their given or known values to perform computations. Substitution is particularly helpful when an equation includes several variables, and we need to find the value of one of them.

In practice, to use the substitution method, you first need to know which values to substitute for which variables. Once you've identified these, you simply replace each variable in the formula with its corresponding value. Here's a step-by-step process that illustrates the procedure using the exercise at hand:

• Identify the variables and their given values: in our case, \( A = 3 \) and \( h = 26 \).
• Locate the formula where the substitution will take place: \( V = A \cdot h \).
• Replace the variables in the formula with the given values: \( V = 3 \cdot 26 \).

By carefully replacing the variables with their actual numbers, you avoid mistakes and can solve the problem effectively.

Multiplication of variables

Multiplication of variables is a pivotal operation in algebra that occurs when you multiply quantities represented by letters. It follows the same principles as multiplying numerical values. When variables are multiplied, their coefficients (the numerical parts) are multiplied together, and the variable parts are theoretically combined. In the case of like terms (the same variable or variables with the same exponents), they are simply aggregated.

For instance, in the problem at hand, we multiply variable A with a numerical value since \( A = 3 \), and we have a variable h with a numerical value of 26. The operation can be expressed as \( 3 \cdot 26 \). Notice that even though one factor is a variable and the other is a number, the multiplication proceeds as usual, leading to the final product of 78 after substitution.

It’s important to grasp that variables act as placeholders for numbers and understanding how to correctly multiply them can allow you to solve a wide range of mathematical problems.