Problem 90 - Algebra A Combined Function

Question

Substitute the given values into each given formula and solve for the unknown variable. If necessary, round to one decimal place. $$ V=\frac{1}{3} \pi r^{2} h ; \quad V=565.2, r=6 $$

Step-by-Step Solution

Answer

The height \( h \) is approximately 15.0 units.

Step 1: Write down the formula

We start with the formula for the volume of a cone: \( V = \frac{1}{3} \pi r^2 h \). This formula calculates the volume \( V \) of a cone given its radius \( r \) and height \( h \).

Step 2: Substitute the given values

We substitute the known values into the formula: \( V = 565.2 \) and \( r = 6 \). This gives us: \[ 565.2 = \frac{1}{3} \pi (6)^2 h \].

Step 3: Simplify and solve for height

First, we simplify the equation: \( 565.2 = \frac{1}{3} \pi \times 36 \times h \). Simplifying further, we have: \( 565.2 = 12\pi h \). To solve for \( h \), divide both sides by \( 12\pi \): \[ h = \frac{565.2}{12\pi} \].

Step 4: Calculate the height

Finally, compute the value of \( h \) using a calculator. The approximate value of \( h \) is \( 565.2 \div (12 \times 3.14159) \approx 15.0 \). We round to one decimal place to get \( h \approx 15.0 \).

Key Concepts

Substitution in Formulas

Substitution is an essential concept when working with algebraic formulas. It involves replacing variables with their corresponding numerical values. This process allows us to solve or simplify equations, making it easier to find unknown variables.

In our exercise about the volume of a cone, we begin by identifying the given values. We have the volume of the cone as 565.2 and the radius as 6. The original formula is:

• Volume (\( V \)) of the cone: \[ V = \frac{1}{3} \pi r^2 h \]

To solve, we substitute the known values into the formula:

• \( V = 565.2 \)
• \( r = 6 \)

This becomes:\[ 565.2 = \frac{1}{3} \pi (6)^2 h \].
By substituting, we've replaced the variables with numbers, allowing us to focus on solving for the remaining unknown variable, which is the height \( h \).

Solving for a Variable

Solving for a variable requires us to find the value of an unknown in an equation. Once substitution is complete, our next step is solving for \( h \).

After substitution, we see:\[ 565.2 = \frac{1}{3} \pi (6)^2 h \].
By rewriting,
this becomes:\[ 565.2 = \frac{1}{3} \times 36 \pi h \].
Further simplifying gives us:\[ 565.2 = 12\pi h \].

To isolate \( h \), divide both sides by \( 12\pi \):\[ h = \frac{565.2}{12\pi} \].
Solving for \( h \) involves performing this division to find its numerical value. Remember, solving for a variable often entails isolating the unknown on one side of the equation, making it possible to calculate its value.

Mathematical Simplification

Mathematical simplification makes equations more manageable. It involves reducing expressions to their simplest form. In our exercise, after substituting, we simplified: \[ 565.2 = \frac{1}{3} \pi (6)^2 h \] to: \[ 565.2 = 12\pi h \].

Simplification can include:

• Eliminating fractions
• Combining like terms
• Reducing coefficients

In this case, simplifying \( \frac{1}{3} \times 36 \pi \) to \( 12\pi \) is crucial. This sets us up to solve for the variable \( h \).

Finally, with \( h = \frac{565.2}{12\pi} \), compute the division using a calculator to obtain:\[ h \approx 15.0 \].
Through simplification, we achieve clearer equations, making it easier to work towards one's final answer.