Problem Solving Questions - Help?

Question

1) Jennifer takes 5.9 hours to drive her car 600 km from kelowna to seattle. for the first part of the trip, her speed was 75 km/h, and for the rest of the trip her speed was 110 km/h. How long did she drive at each speed

2) The perimeter of a lawn is 28 m. The length of hte lawn is 1 m less than twice the width. Determine the length and width of the lawn.

3) At a fast food restauraunt one customer buys three hamburgers and two cheeseburgers for a total of $12.35. Another customer buys five hamburgers and three cheeseburgers for $19.52. Assuming GST is included, determine the cost of each kind of burger

I don't understand the questions, can you perhaps set up how I can solve them? Thank you so much

Answer 1

2) Perimeter of a lawn is the sum of all the sides = 28m

Let x represent the width

The length = 2x - 1

Therefore, perimeter or 28m =  2(length + width)

                                   28 = 2[(2x -1 + x)]

                                   28 = 2(3x -1)

                                   28 = 6x - 2

                                   28 + 2 = 6x

                                   30 = 6x

                                   x= 30/6

                                   x = 5m (width)

To find the length, just plug in value of x in the equation for length = 2x -1

                                                                                                 =2*5-1

                                                                                                = 10-1

                                                                                                  = 9

 

3) Develop equations for the  two scenarios

Let x stand for hamburgers and y for cheeseburgers

1st scenario: 3x + 2y = 12.35

2nd scenario: 5x + 3y = 19.52

Solve for x by process of elimination

In order to 'make' y = zero, multiply the 1st equation by -3 and the 2nd by 2 or the 1st by 3 and the 2nd by -2

3x + 2y = 12.35      multiply by -3

5x + 3y = 19.52      multiply by 2

then, -9x - 6y = -37.05

        10x + 6y= 39.04

Therefore, -9x + 10x - 6y + 6y = -37.05 + 39.04

                  x + 0 = 1.99

                  x = $1.99 ( cost of one hamburger)

To get the cost of a cheeseburger, simply plug in the

value of x into the any of the initial equations: 3x + 2y = 12.35 or 5x + 3y = 19.52

3($1.99) + 2y = 12.35 or

5($1.99) + 3y = 19.52

5.97 + 2y = 12.35 or 9.95 + 3y = 19.52

2y = 12.35-5.98

y = 6.38/2

   = $3.19 9cost of one cheeseburger)

or 3y = 19.52 - 9.95

     3y = 9.57

      y= $3.19 (cost of one cheeseburger).

 

            

Answer 2

Hey Maralyn -- here's more of a "kitchen table" approach:

2) IF lawn is square, you have a 7x7 ... other sides: 8x6, 9x5 <== 9 is 1 < 2(5)

3) Double the $12.35 order for 6H+4C= $24.70 ... this buys xtra H+C for $5.18 ...

    this means 2H+2C costs $10.36, which is $1.99 less than 3H+2C for $12.35 ... 

    Ans. ==> H= $1.99, C=$3.19

1) Average is nearly 100 km/h, so more of trip is driven at 110

... try 4h @ 110= 440 km leaving 1.9h @ 75 < 160 km left -- 17.5 km too short

... 0.1h gains 11 km while losing 7.5 km or 3.5 km more ... try 0.5h more at 110 ...

try 4.5h @ 110= 495 km leaving 1.4h @ 75= 105 km -- just right ... Best wishes :)

Answer 3

For question one 1 we need to use the fact that distance d is given by the equation d = rt. The problem is saying that the total distance, 600km, is broken into 2 parts (one being the distance he travels going 75 km/h and the other being the distance going 110 km/h) we will call these 2 parts d₁ and d₂.

Using the distance equation for d₁ and d₂ we get d₁ = 75t₁ and d₂ = 110t₂  and since we know the total distance is 600km we can form the following equation: d₁ + d₂ = 75t₁ + 110t₂ = 600.

From here we see we have a problem, there are 2 unknown variables and only one equation. To fix this we use the fact that the total time is 5.9 hours to form the equation t₁ + t₂ = 5.9. Now we 2 unknown variables and 2 equations so we can solve the system for t₁ and t₂ and in turn find d₁ and d₂.

Question 2.

We are given that the perimeter P of a rectangular lawn is 28m. this tells us we need to use the equation for the perimeter of a rectangle which is P = 2L + 2W.

The key to solving this problem is to understand how to interpret the 2nd sentence "the length of the lawn is 1m less than twice the width" this sentence is telling us the second equation needed so to say that the length "IS" they are say the length EQUALS 1m less than twice the width. In terms of an equation this is saying the length L = 2W - 1. Now that we have these 2 equations we can now solve for the L and W.

Question 3.

This is another question we need to set up a system for. I will start this one off by first making H represent the cost of hamburgers and C represent the cost of cheeseburgers. It is often helpful to define your own variable at the beginning of some problems, especially if they don't involve already known equations (like perimeter, area, volume, etc.).

Now to set this problem up we break it up into 2 separate scenarios, one for each customer.

Scenario 1.

Buys 2 cheeseburgers and 3 hamburgers and spends $12.35.

Our equation becomes 2C + 3H = 12.35

Scenario 2.

Buys 3 cheeseburgers and 5 hamburgers and spends $19.52.

This equation becomes 3C + 5H = 19.52.

Now we have the 2 necessary equations so we can solve the problem.

I hope this was helpful, please let me know if you need me to explain anything a little more.

Answer 4

1. The best way to start is to figure out what you need to find out, what information you have to get there, and then visually work your way through. There's 2 parts to the question: 

What you need to find out: How many hours the car drove at each speed - 75 km/h and 110 km/h

What you're given: Total hours the trip took - 5.9

Since we don't have the hours @ each speed, we can fill in variables to visually work through it.

X = 75 km/h and Y = 110 km/h

X + Y = 5.9

The other piece is using the rest of the information they gave. Knowing that distance = speed x time, we can make the formula:

600 km/h = D1 + D2

We can fill in more with the rest of the information given & with the distance formula we know D1 = 75 x time and D2 = 110 x time. 

600 km/h = (75 x Time) + (110 x Time)

We already created a time formula above, X + Y = 5.9 so we can add this to our second one

600 km/h = 75X + 110Y

We can fill in the blanks with our resources by flipping them around. 

If 2 + 3 = 5, then 5 - 2 = 3

Same goes for X + Y = 5.9, that means 5.9 - X = Y. Now we have enough information to add to the formula and work our way through

600 km/h = 75X + 110(5.9 - X)

600 km/h = 75x + 649 - 110X

49=35X

X=1.4

If X=1.4, then to find Y we'll use X + Y = 5.9 again.

1.4 + Y = 5.9

Y = 4.5

She drove 1.4 miles ( X ) at 75 km/h and 4.5 miles ( Y ) at 110 km/h.

2. Start with what you're given again (28m) and map out what you need to find.
Width can = X and Length = Y, so you can start with
X + Y = 28m, but because there's 4 sides (2 lengths and 2 widths) it will be
2X + 2Y = 28
Now the rules tell us that the length = twice the width, minus 1
2X - 1 = Y, so we can plug this into our first given formula as:
2X + 2(2X - 1) = 28
2X + 4X - 2 - 28 = 0
2X + 4X - 30 = 0
6X - 30 = 0
X = 5
Now that we know that X = 5, we can fill it into our original formula to find Y

2(5) + 2Y = 28
10 + 2Y = 28
2Y = 18
Y = 9
The length of the lawn (9m) is 1m less than twice the width (5m).
9 = (5 x 2) - 1

 

Since we walked through the first 2, I'm going to let you try #3 to find out how much a hamburger and a cheeseburger cost. :) Good luck!