WORD PROBLEMS INVOLVING RATIONAL FUNCTIONS

Solve each problem and check all solutions. Answer using a complete sentence.

Problem 1 :

A boat goes 240 miles downstream in the same time it can go 160 miles upstream. The speed of the current is 5 miles per hour. What is the speed of the boat in still water?

Solution :

Distance to be covered in downstream = 240 miles 

Distance to be covered in up stream = 160 miles

Let x be the speed of the boat.

Speed in down stream = x + 5

Speed in up stream = x - 5

Time taken for both is the same.

240/(x + 5) = 160/(x - 5)

Doing cross multiplication, we get

240(x - 5) = 160(x + 5)

240x - 1200 = 160x + 800

240x - 160x = 800 + 1200

80x = 2000

x = 2000/80

x = 25

So, the speed of the boat is 25 miles.

Problem 2 :

A plane flies 910 miles with the wind in the same time it can go 660 miles against the wind. The speed of the plane in still air is 305 miles per hour. What is the speed of the wind?

Solution :

With the wind, it can cover the distance of 910 miles.

Against the wind it can cover the distance of = 660 miles.

Speed of the plane = 305 miles

Speed of wind = x

Speed with the wind = x - 305

Against the wind = x + 305

Time = distance / speed

Time taken with the wind = 910 / (305 + x)

Time taken against the wind = 660 / (305 - x)

910 / (305 + x) = 660 / (305 - x)

910 (305 - x) = 660 (305 + x)

277550 - 910x = 201300 + 660x

910x + 660x = 201300 - 277550

1570x = 76250

x = 76250/1570

x = 48.56 miles per hour.

Problem 3 :

A person swims 11 miles downriver in the same time they can swim 7 miles upriver. The speed of the current is 4 miles per hour. Find the speed of the person in still water.

Solution :

Let x be the speed of person in still water.

Distance covered in down river = 11 miles

Distance covered in upriver = 7 miles

Speed in down river = x + 4

Speed in upriver = x - 4

Time taken for downriver = 11/(x + 4)

Time taken for upriver = 7/(x - 4)

11/(x + 4) = 7/(x - 4)

11(x - 4) = 7(x + 4)

11x - 44 = 7x + 28

11x - 7x = 28 + 44

4x = 72

x = 18

So, the speed of the person is 18 miles per hour.

Problem 4 :

Kent can paint a certain room in 6 hours, but Kendra needs 4 hours to paint the same room. How long does it take them to paint the room if they work together?

Solution :

Kent alone finish his work in 6 hours

Work done by him in 1 hour = 1/6

Kendra alone finish his work in 4 hours

Work done by him in 1 hour = 1/4

Work done by them in hour, if they work together = 1/6 + 1/4

= (4 + 6)/24

=10/24

= 5/12

Required time taken = 12/5

= 2.4 hours.

Problem 5 :

Marco can build a laptop twice as fast as Cliff. Working together, it takes them 5 hours. How long would it have taken Marco working alone?

Solution :

Here in the question they are comparing the speed and time taken to complete the work together is given. Converting them as hours.

Let x be the time taken for Cliff, x/2 will be the time taken for Marco.

One hour work by Cliff = 1/x

One hour work by Marco = 1/(x/2) ==> 2/x

Work done by them in one hour, if they work together.

1/x + 2/x = 1/5

(2 + 1)/x = 1/5

3/x = 1/5

15 = x

Time taken by Marco = x/2 = 15/2 ==> 7.5 hours.

Problem 6 :

If s varies inversely as t² , and s = 10 when t = 2, find s when t is 10.

Solution :

s ∝ t²

s = k (1/t²)

s = k/t² -----(1)

When s = 10 and t = 2, we can figure out the value of k.

10 = k/2²

k = 40

Applying the value of k in (1), we get

s = 40/t²

When t = 10, s = ?

s = 40/10²

s = 40/100

s = 2/5

So, the value of s is 2/5, when t = 10.

Problem 8 :

The time (t) traveled by Delmar in a car varies inversely as rate (r). If Delmar drives at a speed of 80 mph in 12 hours, what will be the time to travel if he drives at 60 mph?

Solution :

t - time and r - speed

t ∝ r

t = k (1/r)

t = k/r -----(1)

When r = 80 and t = 12, we can figure out the value of k.

12 = k/80

k = 12(80)

k = 960

applying the value of k in (1), we get

t = 960/r

When r = 60, t = ?

t = 960/60

t = 16

So, the required number of hours is 16 hours.

Problem 9 :

For a given area of a triangle, the base varies inversely as its height. When the height is 10 in the base is 5 in. Find the base if the height is increased to 20 in.

Solution :

Let b the base and h be the height

b ∝ h

b = k/h

height = 10 inches, base = 5 inches

5 = k/10

k = 50

By applying the value of k, we get b = 50/h

Finding base when h = 20 

b = 50/20

h = 5/2

h = 2.5 inches.