Rate of Speed, Ratio and Area of Rectangular Plots

Question

Question 1:  A Concord Jet travels 80 km/h faster than a Delta jet liner. The Concord takes one hour less than the Delta Jet to travel a journey of 6280 km.

Denoting the speed of the Delta Jet Liner by x km/h:

(a) Write down in terms of x expressions for the time taken by:

(i) The Delta Jet Liner
(ii) The Concord Liner

(b) Form an equation to connect these times and show that it simplifies to x2 + 80x - 502400 = 0

(c) Hence find the speed of both aircrafts to the nearest km/h

Question 2:  The ratio of the prices of two different sheets of glass is 2:5. The total bill for 20 sheets of the cheaper glass and 10 sheets of the more expensive one is $1080. If d dollars represent the cost of one sheet of the cheaper glass, determine

(i) An expression in d for the cost of ONE sheet of the more expensive glass
(ii) The value of d
(iii) The cost of ONE sheet of the more expensive glass

Question 3:  Two rectagular plots are equal in area. The length of the first plot is one and a half times its width. The length of the second plot is 7 m less than three times its width.

(a) Denoting the width of the first plot by x m and the width of the second plot by y m, derive a relation between x and y.

(b) If y = x + 1, calculate the values of x and y.

Many thanks...these 3 problems really baffle me.

Answer 1

Hi Roy,

 

These are complex problems for sure, but we'll take each in turn and go step-by-step, so that all will be clear.

 

Problem #1:

 

Let's begin by stating our given information:

    Distance = 6280 km

    Delta Rate = x km/hr

    Concord Rate = x + 80 km/hr

    Concord Time = Delta Time - 1 hr

 

Part A

 

We are asked to come up with an expression in terms of x for the time of each airliner.

 

We can do this by manipulating the Distance formula to get an expression for Time in terms of Distance and Rate:

    Distance = Rate x Time

    Time = Distance / Rate

 

Now we can plug in our given expressions for Distance and Rate of each airliner to come with expressions for the Time of each airliner:

    (1)  Delta Time = 6280 / x  in hrs

    (2)  Concord Time = 6280 / (x + 80) in hrs

 

Part B

 

We are asked to come up with an equation to connect the expressions for airliner times and to derive our equation as x² + 80x - 502400 = 0:

    Concord Time = Delta Time - 1

    6280 / (x + 80) = (6280 / x) - 1

    6280 / (x + 80) = (6280 / x) - (x / x)

    6280 / (x + 80) = (6280 - x) / x

    (6280)(x) = (6280 - x)(x + 80)

    6280x = 6280x + 502400 - x² - 80x

    6280x - 6280x = 502400 - x² - 80x

    -x² -80x + 502400 = 0

    -1(x² + 80x - 502400) = 0

    x² + 80x - 502400 = 0

 

Part C

 

We are asked to find the speed of each airliner using the equation obtained in Part B:

    x² + 80x - 502400 = 0

 

We'll use the quadratic formula since this equation is NOT factorable:

    ax² + bx + c = 0 (a = 1; b = 80; c= -502400)

    x = (-b +/- √(b² - 4ac)) / 2a

    x = (-80 +/- √(80² - 4(1)(-502400))) / 2(1)

    x = (-80 +/- √(6400 + 2009600)) / 2

    x = (-80 +/- √2016000) /2

    x = (-80 +/- 1420) / 2

    x = -40 - 710 = -750 (reject negative answer)

    x = - 40 + 710 = 670 km/hr (Delta Rate)

    x + 80 = 670 + 80 = 750 km/hr (Concord Rate)

 

Problem #2:

 

Let's begin by stating our given information:
    Cheap Price : Expensive Price = 2 : 5

    Cheap Cost + Expensive Cost = $1080

    d = Cheap Cost of 1 sheet

 

Part I

 

In terms of d what is the Expensive Cost of 1 sheet?       2 / 5 = d / x

     2x = 5d 

      x = (5/2)d (Expensive Cost of 1 sheet)

 

Part II

 

What is the value of d?

    Cheap Cost + Expensive Cost = 1080 

    20d + 10(5d/2) = $1080

    20d + 25d = 1080

             45d = 1080

                d  = 1080/45

                d = $24 (Cheap Cost of 1 sheet)

 

Part III

 

What is the Expensive Cost of 1 sheet?

    Expensive Cost of 1 sheet = 5d/2

                                         = 5(24)/2

                                         = 120/2

                                         = $60

 

Problem #3:

 

Let's begin by stating our given information:
    Rectangular Plot 1 = Rectangular Plot 2

    Rectangular Plot 1:

        L = 1.5x

    Rectangular Plot 2:

        L = 3y - 7

 

Part A

 

What is the relation between x and y?

    Area 1 = Area 2

    (3/2)(x)(x) = (3y - 7)(y)

    (3/2)x² = 3y² - 7y

 

Part B

 

If y = x +1, calculate values of x and y:

    (3/2)x² = 3y² - 7y

     (3/2)x^{2 =} 3(x + 1)² - 7(x + 1)

    (3/2)x² = 3(x² + 2x + 1) - 7x - 7

    (3/2)x² = 3x² + 6x + 3 - 7x - 7

    (3/2)x² = 3x² - x - 4

    3x² = 6x² - 2x - 8

    3x² - 2x - 8 = 0

    (3x + 4)(x - 2) = 0

    3x + 4 = 0      |  x - 2 = 0

          3x = -4     |  x = 2

            x = -4/3  |  x = 2  (accept positive answer)

            y = x + 1 = 2 + 1 = 3 

 

Whew - lots of work for sure !!

 

Thanks for submitting these problems & glad to help.

 

God bless, Jordan.