Q53. - Algebra 1

Question

The Red Rocks Amphitheater located in the Red Rock Park near Denver, Colorado, is the only naturally occurring amphitheater. The acoustic qualities here are such that a maximum of 20,000 people, plus or minus 1000 can hear natural voices clearly.

1. Write an equation involving an absolute value that represents the number of people that can hear natural voices at Red Rocks Amphitheater.
2. Find the maximum and minimum number of people that can hear natural voices clearly in the amphitheater.
3. What is the range of people in part b?

Step-by-Step Solution

Answer

1. The absolute value equation that represents time $\left|t-20000\right|=1000$.
2. The maximum number of people are 21000 and the minimum number of people are 19000.
3. The range of people is $\left(19000, 21000\right)$.

Part a. Step 1. Apply the concept of Variables and constant.

Variables: - In any expression or equation, the variable is an unknown quantity. It may be any alphabet or any special symbol.

Constants: - In any expression or equation constant are the values which is fixed.

While performing addition in an equation, same kind of elements or like terms are added together.

Part a. Step 2. Concept of absolute value.

Absolute value: - Absolute value of any quantity simply means positive value of that quantity.

Example: -

Consider a variable x. Here x can be both positive or negative but $\left|x\right|$ will always be positive and hence the absolute value. 

Part a. Step 3. Solution along with the given information.

It is given that the acoustic qualities here are maximum of 20000 people, plus or minus 1000 can hear natural voice clearly

Consider the expression.

The recording time be t.

Therefore, the absolute value equation that represents time 

$\left|t-20000\right|=1000$ .

Part b. Step 1. Apply the concept of Variables and constant.

Variables: - In any expression or equation, the variable is an unknown quantity. It may be any alphabet or any special symbol.

Constants: - In any expression or equation constant are the values which is fixed.

While performing addition in an equation, same kind of elements or like terms are added together.

Part b. Step 2. Concept of absolute value.

Absolute value: - Absolute value of any quantity simply means positive value of that quantity.

Example: -

Consider a variable x. Here x can be both positive or negative but $\left|x\right|$ will always be positive and hence the absolute value.

Part b. Step 3. Solution along with the given information.

It is given that the acoustic qualities here are maximum of 20000 people, plus or minus 1000 can hear natural voice clearly

Consider the expression.

The recording time be t.

Therefore, the absolute value equation that represents time 

$\left|t-20000\right|=1000$.

 

Case 1. $\left(t-20000\right)=1000$

Add 20000 to both the sides of the equation.

$\begin{array}{l}t-20000+20000=1000+20000\\ t-0=21000\\ t=21000\end{array}$ 

 

Case 2. $\left(t-20000\right)=-1000$

Add 90 to both the sides of the equation.

$\begin{array}{l}t-20000+20000=-1000+20000\\ t-0=19000\\ t=19000\end{array}$ 

 

The maximum number of people are 21000 and the minimum number of people are 19000.

Part c. Step 1. Apply the concept of Variables and constant.

Variables: - In any expression or equation, the variable is an unknown quantity. It may be any alphabet or any special symbol.

Constants: - In any expression or equation constant are the values which is fixed.

While performing addition in an equation, same kind of elements or like terms are added together.

Part c. Step 2. Concept of absolute value.

Absolute value: - Absolute value of any quantity simply means positive value of that quantity.

Example: -

Consider a variable x. Here x can be both positive or negative but $\left|x\right|$ will always be positive and hence the absolute value. 

Part c. Step 3. Range.

Range is simply the interval in between which the values of the solution falls

Here the maximum number of people are 21000 and the minimum number of people are 19000.

So, the range is $\left(19000, 21000\right)$.