Jane rowed her canoe against a 1-mile-per-hour current upstream 12 miles and then returned the 12 miles back downstream. If the total trip took 5 hours, then at what speed can Jane row in still water? Jose drove 15 miles to pick up his sister and then returned home. On the return trip, he was able to average 15 miles per hour faster than he did on the trip to pick her up. Barry drove the 24 miles to town and then back in 1 hour. On the return trip, he was able to average 14 miles per hour faster than he averaged on the trip to town.
What was his average speed on the trip to town? Jerry paddled his kayak upstream against a 1-mile-per-hour current for 12 miles. The return trip downstream with the 1-mile-per-hour current took 1 hour less time. How fast can Jerry paddle the kayak in still water? It takes a light aircraft 1 hour more time to fly miles against a mile-per-hour headwind than it does to fly the same distance with it. What is the speed of the aircraft in calm air? James can paint the office by himself in 7 hours.
Manny paints the office in 10 hours. How long will it take them to paint the office working together? Barry can lay a brick driveway by himself in 12 hours. Robert does the same job in 10 hours. How long will it take them to lay the brick driveway working together? Jerry can detail a car by himself in 50 minutes.
Sally does the same job in 1 hour. How long will it take them to detail a car working together? Jose can build a small shed by himself in 26 hours. Alex builds the same small shed in 2 days. How long would it take them to build the shed working together? Allison can complete a sales route by herself in 6 hours. Working with an associate, she completes the route in 4 hours. How long would it take her associate to complete the route by herself?
James can prepare and paint a house by himself in 5 days. Working with his brother, Bryan, they can do it in 3 days. How long would it take Bryan to prepare and paint the house by himself? Joe can assemble a computer by himself in 1 hour. Working with an assistant, he can assemble a computer in 40 minutes. How long would it take his assistant to assemble a computer working alone? If the teacher helps, then the grading can be completed in 20 minutes.
How long would it take the teacher to grade the papers working alone? A larger pipe fills a water tank twice as fast as a smaller pipe. When both pipes are used, they fill the tank in 5 hours. If the larger pipe is left off, then how long would it take the smaller pipe to fill the tank? A newer printer can print twice as fast as an older printer. If both printers working together can print a batch of flyers in 45 minutes, then how long would it take the newer printer to print the batch working alone?
Working alone, Henry takes 9 hours longer than Mary to clean the carpets in the entire office. Working together, they clean the carpets in 6 hours. How long would it take Mary to clean the office carpets if Henry were not there to help? Working alone, Monique takes 4 hours longer than Audrey to record the inventory of the entire shop. Working together, they take inventory in 1.
How long would it take Audrey to record the inventory working alone? Jerry can lay a tile floor in 3 hours less time than Jake. If they work together, the floor takes 2 hours. How long would it take Jerry to lay the floor by himself? Jeremy can build a model airplane in 5 hours less time than his brother. Working together, they need 6 hours to build the plane. How long would it take Jeremy to build the model airplane working alone?
Harry can paint a shed by himself in 6 hours. Jeremy can paint the same shed by himself in 8 hours. How long will it take them to paint two sheds working together? Joe assembles a computer by himself in 1 hour.
Working with an assistant, he can assemble 10 computers in 6 hours. How long would it take his assistant to assemble 1 computer working alone? Jerry can lay a tile floor in 3 hours, and his assistant can do the same job in 4 hours. If Jerry starts the job and his assistant joins him 1 hour later, then how long will it take to lay the floor?
Working alone, Monique takes 6 hours to record the inventory of the entire shop, while it takes Audrey only 4 hours to do the same job. How long will it take them working together if Monique leaves 2 hours early? Consider a freight train moving at a constant speed of 30 miles per hour. The equation that expresses the distance traveled at that speed in terms of time is given by.
After 1 hour the train has traveled 30 miles, after 2 hours the train has traveled 60 miles, and so on. We can construct a chart and graph this relation. In this example, we can see that the distance varies over time as the product of the constant rate, 30 miles per hour, and the variable, t. In this form, it is reasonable to say that D is proportional to t , where 30 is the constant of proportionality. In general, we have. Here k is nonzero and is called the constant of variation The nonzero multiple k , when quantities vary directly or inversely.
If the circumference is measured to be 20 inches, then what is the radius of the circle? Therefore, we write. Now use this formula to find d when the circumference is 20 inches. The radius of the circle, r , is one-half of its diameter. Typically, we will not be given the constant of variation. Instead, we will be given information from which it can be determined. If a man weighs pounds on earth, then he will weigh 30 pounds on the moon.
Set up an algebraic equation that expresses the weight on earth in terms of the weight on the moon and use it to determine the weight of a woman on the moon if she weighs pounds on earth. To find the constant of variation k , use the given information.
Solve for k. Next, set up a formula that models the given information. Answer: The woman weighs 20 pounds on the moon. Next, consider the relationship between time and rate,. If we wish to travel a fixed distance, then we can determine the average speed required to travel that distance in a given amount of time.
For example, if we wish to drive miles in 4 hours, we can determine the required average speed as follows:. The average speed required to drive miles in 4 hours is 60 miles per hour.
If we wish to drive the miles in 5 hours, then determine the required speed using a similar equation:. In this case, we would only have to average 48 miles per hour. We can make a chart and view this relationship on a graph. This is an example of an inverse relationship.
We say that r is inversely proportional to the time t , where is the constant of proportionality. Again, k is nonzero and is called the constant of variation or the constant of proportionality. Therefore, the formula that models the problem is. Example 4: The weight of an object varies inversely as the square of its distance from the center of earth.
Use the given information to find k. An object weighs pounds on the surface of earth, approximately 4, miles from the center. Therefore, we can model the problem with the following formula:. To use the formula to find the weight, we need the distance from the center of earth. Since the object is 1, miles above the surface, find the distance from the center of earth by adding 4, miles:. Answer: The object will weigh 64 pounds at a distance 1, miles above the surface of earth.
Lastly, we define relationships between multiple variables. Here k is nonzero and is called the constant of variation or the constant of proportionality. Give a formula for the area of an ellipse. Therefore, the formula for the area of an ellipse is. Translate the following sentences into a mathematical formula. The distance, D , an automobile can travel is directly proportional to the time, t , that it travels at a constant speed.
The extension of a hanging spring, d , is directly proportional to the weight, w , attached to it. The volume, V , of a sphere varies directly as the cube of its radius, r. The volume, V , of a given mass of gas is inversely proportional to the pressure, p , exerted on it.
The intensity, I , of light from a light source is inversely proportional to the square of the distance, d , from the source. Every particle of matter in the universe attracts every other particle with a force, F , that is directly proportional to the product of the masses, m 1 and m 2 , of the particles and inversely proportional to the square of the distance, d , between them. Simple interest, I , is jointly proportional to the annual interest rate, r , and the time, t , in years a fixed amount of money is invested.
The period, T , of a pendulum is directly proportional to the square root of its length, L. The time, t , it takes an object to fall is directly proportional to the square root of the distance, d , it falls. Construct a mathematical model given the following.
Applications involving variation. Revenue in dollars is directly proportional to the number of branded sweat shirts sold. The sales tax on the purchase of a new car varies directly as the price of the car. The price of a share of common stock in a company is directly proportional to the earnings per share EPS of the previous 12 months. The distance traveled on a road trip varies directly with the time spent on the road.
If a mile trip can be made in 3 hours, then what distance can be traveled in 4 hours? The circumference of a circle is directly proportional to its radius. The area of circle varies directly as the square of its radius. The surface area of a sphere varies directly as the square of its radius. Find the surface area of a sphere with radius 3 meters. The volume of a sphere varies directly as the cube of its radius. Find the volume of a sphere with radius 1 meter.
With a fixed height, the volume of a cone is directly proportional to the square of the radius at the base. When the radius at the base measures 10 centimeters, the volume is cubic centimeters.
Determine the volume of the cone if the radius of the base is halved. The distance, d , an object in free fall drops varies directly with the square of the time, t , that it has been falling. If an object in free fall drops 36 feet in 1. The constant of variation is called the spring constant. If a hanging spring is stretched 3 centimeters when a 2-kilogram weight is attached to it, then determine the spring constant. If a hanging spring is stretched 6 centimeters when a 4-kilogram weight is attached to it, then how far will it stretch with a 2-kilogram weight attached?
The breaking distance of an automobile is directly proportional to the square of its speed. If it takes 36 feet to stop a particular automobile moving at a speed of 30 miles per hour, then how much breaking distance is required if the speed is 35 miles per hour?
After an accident, it was determined that it took a driver 80 feet to stop his car. In an experiment under similar conditions, it takes 45 feet to stop the car moving at a speed of 30 miles per hour.
Estimate how fast the driver was moving before the accident. A balloon is filled to a volume of cubic inches on a diving boat under 1 atmosphere of pressure. If the balloon is taken underwater approximately 33 feet, where the pressure measures 2 atmospheres, then what is the volume of the balloon?
If a balloon is filled to cubic inches under a pressure of 3 atmospheres at a depth of 66 feet, then what would the volume be at the surface, where the pressure is 1 atmosphere? To balance a seesaw, the distance from the fulcrum that a person must sit is inversely proportional to his weight. If a pound boy is sitting 3 feet from the fulcrum, then how far from the fulcrum must a pound boy sit to balance the seesaw?
The current, I , in an electrical conductor is inversely proportional to its resistance, R. The number of men, represented by y , needed to lay a cobblestone driveway is directly proportional to the area, A , of the driveway and inversely proportional to the amount of time, t , allowed to complete the job. Typically, 3 men can lay 1, square feet of cobblestone in 4 hours. How many men will be required to lay 2, square feet of cobblestone given 6 hours?
The volume of a right circular cylinder varies jointly as the square of its radius and its height. Find a formula for the volume of a right circular cylinder in terms of its radius and height. If the length of a pendulum is 1 meter, then the period is approximately 2 seconds.
Approximate the period of a pendulum that is 0. How long will it take an object dropped from 16 feet to hit the ground? The constant of proportionality is called the gravitational constant. Calculate the gravitational constant.
Use the gravitational constant from the previous exercise to write a formula that approximates the force, F , in newtons between two masses m 1 and m 2 , expressed in kilograms, given the distance d between them in meters.
Calculate the force in newtons between earth and the moon, given that the mass of the moon is approximately 7. Calculate the force in newtons between earth and the sun, given that the mass of the sun is approximately 2. If y varies directly as the square of x , then how does y change if x is doubled? If y varies inversely as square of t , then how does y change if t is doubled? If y varies directly as the square of x and inversely as the square of t , then how does y change if both x and t are doubled?
State the restrictions and simplify. Mary can jog, on average, 2 miles per hour faster than her husband, James. James can jog 6. How fast, on average, can Mary jog? Billy traveled miles to visit his grandmother on the bus and then drove the miles back in a rental car. The bus averages 14 miles per hour slower than the car. If the total time spent traveling was 4. Jerry takes twice as long as Manny to assemble a skateboard. If they work together, they can assemble a skateboard in 6 minutes.
Working alone, Joe completes the yard work in 30 minutes. It takes Mike 45 minutes to complete work on the same yard. How long would it take them working together? The distance an object in free fall drops varies directly with the square of the time that it has been falling. It is observed that an object falls 16 feet in 1 second.
Find an equation that models the distance an object will fall and use it to determine how far it will fall in 2 seconds. The weight of an object varies inversely as the square of its distance from the center of earth. Simplify and state the restrictions. Assume all variables in the denominator are positive.
Set up an algebraic equation and then solve. An integer is three times another. Working alone, Joe can paint the room in 6 hours. If Manny helps, then together they can paint the room in 2 hours. How long would it take Manny to paint the room by himself? A river tour boat averages 6 miles per hour in still water. With the current, the boat can travel 17 miles in the same time it can travel 7 miles against the current.
Under optimal conditions, a certain automobile moving at 35 miles per hour can break to a stop in 25 feet. Find an equation that models the breaking distance under optimal conditions and use it to determine the breaking distance if the automobile is moving 28 miles per hour. Previous Chapter. Table of Contents. Next Chapter. Chapter 7 Rational Expressions and Equations. Simplify rational expressions. Simplify expressions with opposite binomial factors.
Simplify and evaluate rational functions. Simplifying Rational Expressions When simplifying fractions, look for common factors that cancel. Video Solution click to see video. Rational Functions Rational functions have the form. Key Takeaways Rational expressions usually are not defined for all real numbers. The real numbers that give a value of 0 in the denominator are not part of the domain. These values are called restrictions. Simplifying rational expressions is similar to simplifying fractions.
First, factor the numerator and denominator and then cancel the common factors. Rational expressions are simplified if there are no common factors other than 1 in the numerator and the denominator.
Simplified rational expressions are equivalent for values in the domain of the original expression. Be sure to state the restrictions if the denominators are not assumed to be nonzero. Use the opposite binomial property to cancel binomial factors that involve subtraction. Fill in the following chart:. Divide rational expressions. Rational expressions and rational exponents are both basic mathematical constructs used in a variety of situations. Both types of expressions can be represented both graphically and symbolically.
The most general similarity between the two is their forms. A rational expression and a rational exponent are both in the form of a fraction. Their most general difference is that a rational expression is composed of a polynomial numerator and denominator. A rational exponent can be a rational expression or a constant fraction. In the sciences, rational expressions are used as simplified models of complex equations in order to more easily approximate results without requiring time-consuming complex math.
Rational expressions are commonly used to describe phenomena in sound design, photography, aerodynamics, chemistry and physics. Unlike rational exponents, a rational expression is an entire expression, not just a component. The graphs of most rational expressions are discontinuous, meaning they contain a vertical asymptote at certain values of x that are not part of the domain of the expression.
Once you start looking at graphs of rational functions, those "canceled" factors are what lead to holes in the graph. Factor the denominator.
If you are faced with the sum or difference of two rational expressions, find the least common denominator. Multiply each term by "1" to make the denominators the same, combine like terms in the numerator, and simplify. In order to find the least common denominator, factor both denominators. Now you have something that looks more complicated, but the denominators are the same, so we'll put it all together over a single denominator.
When dealing with rational expressions, your "answer" will still probably be a rational expression, just a simpler one. This is a rational equation, and nothing can be factored further at this point. You can multiply both sides of the equation by the common denominator, and when you do, factors will for many problems cancel. We have left rational expressions and are back to friendly polynomials. Since it's a quadratic now, move everything to one side. Always double check that the solutions work.
When checking rational functions, make sure they don't create a zero in one of the original denominators. When checking solutions to radical functions, completely check each solution you find.
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