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Chapter 5: Problem 16

You and your family are visiting Seattle and take the elevator to the observation deck of the Space Needle. The observation deck is \(520 \mathrm{ft}\) high while the needle itself is \(605 \mathrm{ft}\) high. The elevator travels at a constant speed, and it takes \(43 \mathrm{~s}\) to travel from the base at \(0 \mathrm{ft}\) to the observation deck. a. What is the slope of the graph of this situation? (a) b. If the elevator could go all the way to the top, how long would it take to get there? c. If a rider got on the elevator at the restaurant at the \(100 \mathrm{ft}\) level, what equation models her ride to the observation deck? (a)

Short Answer

Expert verified
(a) The slope is 12.09 ft/s. (b) It takes ~50.08 s to reach the top. (c) The model is 34.74 s from 100 ft to 520 ft.

Step by step solution

01

Identify Information for Slope

The slope is the speed of the elevator, which can be calculated by using the change in height over the change in time. Here, the height changes from 0 ft to 520 ft in 43 seconds.
02

Calculate the Slope

The formula for the slope is \( m = \frac{\text{change in height}}{\text{change in time}} \). Substituting the given numbers, we have \( m = \frac{520 - 0}{43} = \frac{520}{43} \approx 12.09 \mathrm{ft/s} \).
03

Time to the Top

We need to find how long it would take to travel 605 ft using the same speed. Using the formula \( t = \frac{d}{v} \), where \( d = 605 \mathrm{ft} \) and \( v = 12.09 \mathrm{ft/s} \), we calculate: \( t = \frac{605}{12.09} \approx 50.08 \mathrm{~s} \).
04

Write Equation for Specific Ride

To find the time for a ride starting from 100 ft to 520 ft, use the equation of a line: \( h = mt + b \), where \( h \) is the height. Here, the slope \( m \) is 12.09 ft/s, \( h = 520 \) when \( t = 0 \) (from 100 ft not ground level), so the equation is \( h = 12.09t + 100 \). Solve for \( t \) when \( h = 520 \), thus \( 520 = 12.09t + 100 \) which simplifies to \( t = \frac{420}{12.09} \approx 34.74 \mathrm{s} \).

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Key Concepts

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

Slope Calculation
Calculating the slope is a fundamental concept in algebra that's not only vital for solving this particular exercise but serves as a key skill in various mathematical applications. In this scenario, the slope essentially represents the speed at which the elevator travels. The slope formula is given by \[ m = \frac{\text{change in height}}{\text{change in time}} \]. This formula helps us determine how much the elevator ascends per second.
  • The change in height is between the two points you're interested in 鈥� here, from 0 to 520 feet.
  • The change in time is how long it takes 鈥� 43 seconds in this case.
By substituting the values into the formula, we get the slope, or speed of the elevator, which is approximately 12.09 feet per second. This allows us a concrete measure to understand how swiftly the elevator is moving relative to time.
Linear Equation
A linear equation is a type of equation that results in a straight line when graphed on a coordinate plane. It generally takes the form \[ y = mx + b \], where
  • \( m \) represents the slope or rate of change, showing how steep the line is.
  • \( b \) is the y-intercept, which is the value of \( y \) when \( x = 0 \).
In this elevator problem, the linear equation models the movement of the elevator in feet over time in seconds. For a ride starting at the restaurant located at 100 feet, we use the equation \[ h = 12.09t + 100 \]. Here, the slope of 12.09 symbolizes the elevator's speed, and the 100 denotes the starting height. By inputting different values of \( t \), or time, we can predict the elevator's height at any given moment.
Rate of Change
The concept of rate of change is pivotal in understanding how one quantity shifts with another over time. In many mathematical scenarios, it's essential to comprehend whether something increases or decreases steadily, or fluctuates in a non-linear fashion. In our problem, the rate of change is the speed of the elevator, which is evenly shifting upwards at a rate of 12.09 feet per second.
  • This is a constant rate of change, meaning it remains the same no matter what interval of time we consider.
  • In real-life applications, such consistency is crucial for planning and safety considerations, such as estimating travel time in buildings.
Understanding this concept helps students not only apply it in exercises like this but also see its practical use in everyday situations, providing a strong foundation for more advanced studies in math and science.

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Most popular questions from this chapter

APPLICATION Each day, Sal prepares a large basket of self-serve tortilla chips in his restaurant. On Monday, 40 adult patrons and 15 child patrons ate \(10.8 \mathrm{~kg}\) of chips. On Tuesday, 35 adult patrons and 22 child patrons ate \(12.29 \mathrm{~kg}\) of chips. Sal wants to know whether adults or children eat more chips on average. a. Organize the information into a table. (a) b. Define variables and write a system of equations. (a) c. Write a matrix for the system. d. Solve the system by transforming the matrix into the solution matrix \(\left[\begin{array}{lll}1 & 0 & a \\ 0 & 1 & b\end{array}\right]\). e. Write a sentence that describes the real-world meaning of the solution to the system.

Consider this system of equations: $$ \left\\{\begin{array}{l} 2 x-5 y=12 \\ 6 x-15 y=36 \end{array}\right. $$ a. By what number can you multiply which equation to eliminate the \(x\)-term when you combine the equations by addition? Do this multiplication. b. What is the sum of these equations? c. What is the solution to the system? d. How can you predict this result by examining the original equations?

On Kids' Night, every adult admitted into a restaurant must be escorted by at least one child. The restaurant has a maximum seating capacity of 75 people. a. Write a system of inequalities to represent the constraints in this situation. (a) b. Graph the solution. Is it possible for 50 children to escort 10 adults into the restaurant? c. Why might the restaurant reconsider the rules for Kids' Night? Add a new constraint to address these concerns. Draw a graph of the new solution.

At the Coffee Stop, you can buy a mug for \(\$ 25\) and then pay only \(\$ 0.75\) per hot drink. a. What is the slope of the equation that models the total cost of refills? What is the real-world meaning of the slope? b. Use the point \((33,49.75)\) to write an equation in point-slope form that models this situation. c. Rewrite your equation in intercept form. What is the real-world meaning of the \(y\)-intercept?

Part of Adam's homework paper is missing. If \((5,2)\) is the only solution to the system shown, write a possible equation that completes the system. (Ti)
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