Equations,money problems, graphs-help please

Question

Before hiring a car for a day trip, Kyle called several car hire services for some quotes.

Company:                         Cost Per day:

Sam’s Car Hire                  $25, plus $0.07 (7 cents) for each km.

Fleet Car Hire                    $50 for each day.

Daytripper’s Car Hire        $35, plus $0.05 (5 cents) for each km.

1.       Set yp a table of values for each car hire firm showing trips from 0km to 450km in 50km increments.

 

Let K stand for number of kilometres travelled

Let Cs stand for Sam’s hire cost

Let Cf stand for Fleet hire cost

Let Cd stand for Daytripper’s hire cost

 

a)      For each company write an equation for the cost.

b)      Plot three equations on the same graph.

Let the x axis have the distance with values from 0 to 600km, and the y axis show the cost C with values from 0 to $80.

c)       Explain which deal is better for

(i)                  Short trips ?

(ii)                Long trips?

Sometimes it makes no difference which company you use.

d)      For what distances does:

(i)                  Cs=Cf?

(ii)                Cf=Cd?

(iii)               Cs=Cd?

 

                   2.    For the design below: 

                    http://www.flickr.com/photos/99973477@N03/11444159853/

a)      Draw the next three shapes in the sequence.

b)      Draw up a table for the first ten shapes in the sequence, and the number of matches required to complete the shape.

c)       Determine the word relationship for the sequence of outputs.

d)      Determine the formula which would allow you to find the number of matches in any shape in the sequence.

e)      Draw a graph of the relationship.

 

3.       For the design below:

                         http://www.flickr.com/photos/99973477@N03/11444055415/

a)      Draw the next three shapes in the sequence.

b)      Draw up a table for the first ten shapes in the sequence, and the number of matches required to complete the shape.

c)       Determine the word relationship for the sequence of outputs.

d)      Determine the formula which would allow you to find the number of matches in any shape in the sequence.

e)      Draw a graph of the relationship.

 

4. A student banks $200 when she starts secondary College in Grade 7. In Grade 8 she banks $250,$300 in Grade 9, and so on. She is at school for 6 years.

a)  Determine a formula relating the amount deposited during a year (200,250,...) to how many years they have been saving(1,2,...).

Because the deposits are increasing, the formula for the bank balance is more complex and is not a linear relationship. Answer the following using a table.

b)  What is the total amount she has banked by the time she finishes after 6 years?

c)   If she continued banking in this manner after she leaves school, how many years(from the start) will it take for her balance to exceed $15,000?

 

Answer 1

1) kyle called several car hire services

k	Cs	Cf	Cd
0	25	50	35
50	28.5	50	37.5
100	32	50	40
150	35.5	50	42.5
200	39	50	45
250	42.5	50	47.5
350	46	50	50
400	49.5	50	52.5
450	53	50	55

a)Equation for the each company for the cost

Cs cost is y = 25+0.07x

Cf cost is y = 50

Cd cost is y = 35+0.05x

Comments

Above solution (1) is wrong.

See the correct solution below.

Answer 2

b) Three equations on same graph.

Draw the coordinate plane.

Let the x axis have the distance with values from 0 to 600km and the y axis show the cost C with values from 0 to $80.

From the  equations.

 y = 25+0.07x

 y = 50

 y = 35+0.05x

Answer 3

c) Sam’s Car Hire company deal is better for short trips, why because Compare to other two companies

From 0 to 350 km price is lowest.

Fleet Car Hire company deal is better for long tripes, why because compare to other two companies 

it is maintain standard price to short or long distance . It's cost only $50 for each day.

Answer 4

1) kyle called several car hire services

k	Cs	Cf	Cd
0	25	50	35
50	28.5	50	37.5
100	32	50	40
150	35.5	50	42.5
200	39	50	45
250	42.5	50	47.5
300	46	50	50
350	49.5	50	52.5
400	53	50	55
450	56.5	50	57.5

Answer 5

a)Equation for the each company for the cost

Cs cost is y = 25+0.07x

Cf cost is y = 50

Cd cost is y = 35+0.05x

Answer 6

C) Sometimes it makes no difference i prefer fleet car hire company because i think it gives comfortable journey compare remaining company cars.

d) (i) when 350 km, Cs cost is 49.5 and Cf cost is 50,it seems to be equal.(approximately)

(ii) when distance is 300 km, Cf equals to Cd.

Cost is 50.

(iii) When distance is 500 km, Cs equls to Cd.

Cost is 60

Answer 7

(3).(a).

 

(b). Make the table

Numbers	1	2	3	4	5	6	7	8	9	10
Number of shapes	1	4	7	10	13	16	19	22	25	28

 

(c). The word relationship for the sequence of outputs :

Number of shapes = The difference between three times a number(n) and two.

 

(d). Determine the formula which would allow you to find the number of matches in any shape in the sequence.

Sequence : 1, 4, 7, 10, . . . . . . .nth term.

Since common difference is 3, the above sequence is arithmetic sequence.

nth term t_n = a₁ + (n - 1)d, where a₁ = first term and d = common term.

t_n = ₁ + (n - 1)3

t_n = 1 + 3n - 3

t_n = 3n - 2.

 

(e). Draw a graph of the relationship y = 3x - 2.

 

Answer 8

(2).(a). The next three shapes in the sequence :

 

(b). Make the table

Numbers	1	2	3	4	5	6	7	8	9	10
Number of shapes	1	2	3	4	5	6	7	8	9	10

 

(c). The word relationship for the sequence of outputs :

Number of shapes = number(n).

 

(d). Determine the formula which would allow you to find the number of matches in any shape in the sequence.

Sequence : 1, 2, 3, 4, . . . . . . .n.

Since common difference is 1, the above sequence is arithmetic sequence.

nth term t_n = a₁ + (n - 1)d, where a₁ = first term and d = common term.

t_n = 1 + (n - 1)1

t_n = 1 + n - 1

t_n = n.

 

(e). Draw a graph of the relationship y = x.

 

Answer 9

(4).(a).

Year                     Balance after each compoundig.

  0                        200                            [P = P]

  1                        200 + 250 = 450        [P₁ = P + Pr = P(1 + r)]

  2                        450 + 300 = 750        [P₂ = P₁(1 + r) = P(1 + r)(1 + r) = P(1 + r)²]

  3                        750 + 350 = 1100      [P₃ = P₂(1 + r) = P(1 + r)²(1 + r) = P(1 + r)³]

  .                        .                                             .

  .                        .                                             .

  .                        .                                             .

  t                        .                                [P_t = P(1 + r)^t]

For n compounings per year : A = P(1 + r/n)^nt, where P = principal, r = interest rate(in decimal form), t = number of years.

For continuous compouning : A = Pe^rt.

 

(b). The total amount after 6 years :

Here find the r value :

If A = 450, P = 200 and t = 1 then 450 = 200e^(1)r    ------> (1).

If A = 750, P = 200 and t = 2 then 750 = 200e^(2)r    ------> (2).

Divide the equation 2 by equation 1.

750/450 = 200e^2r / 200e^r -----> e^r = 1.67  -----> r = ln 1.67 = 0.5.

The total amount after 6 years A = (200)e^(0.5)(6) = 200 * 20.0855 = 4017.107 dollars.

 

(C). Find the time in years :

A = 15, 000 dollars and r = 0.5, P = 200.

A = Pe^rt.

15,000 = (200)e^(0.5)(t)

75 = e^t/2 =

ln 75 = t/2

8.635 years = t.

Answer 10

(1). Make the table :

k	Cs = 25 + 0.07k	Cf = 50	Cd = 35 + 0.05k
0	Cs = 25 + 0.07(0) = 0	50	Cd = 35 + 0.05(0) = 35
50	Cs = 25 + 0.07(50) = 28.5	50	Cd = 35 + 0.05(50) = 37.5
100	Cs = 25 + 0.07(100) = 32	50	Cd = 35 + 0.05(100) = 40
150	Cs = 25 + 0.07(150) = 35.5	50	Cd = 35 + 0.05(150) = 42.5
200	Cs = 25 + 0.07(200) = 39	50	Cd = 35 + 0.05(200) = 45
250	Cs = 25 + 0.07(250) = 42.5	50	Cd = 35 + 0.05(250) = 47.5
300	Cs = 25 + 0.07(300) = 46	50	Cd = 35 + 0.05(300) = 50
350	Cs = 25 + 0.07(350) = 49.5	50	Cd = 35 + 0.05(350) = 52.5
400	Cs = 25 + 0.07(400) = 53	50	Cd = 35 + 0.05(400) = 55
450	Cs = 25 + 0.07(450) = 56.5	50	Cd = 35 + 0.05(450) = 57.5

 

(a).

Company name                          Equation for the cost

Sam’s Car Hire                           C_s = 25 + 0.07k

Fleet Car Hire                             C_f = 50

Daytripper’s Car Hire                  C_d = 35 + 0.05k.

 

(b). Plot three equations on the same graph.

 

 

C).

(i). From the graph, Sam’s Car Hire company deals is better for Short trips.

(ii). From the graph, Fleet Car Hire company deals is better for Long trips.

 

(d).

(i). C_s = C_f

⇒ 25 + 0.07k = 50

⇒ 0.07k = 25

⇒ k = 357.143 km.

 

(ii). C_f = C_d

⇒ 50 = 35 + 0.05k

⇒ 15 = 0.05k

⇒ k = 300 km

 

(iii). C_s = C_d C_f = 50 C_d = 35 + 0.05k.

⇒ 25 + 0.07k = 35 + 0.05k

⇒ 0.02k = 10

⇒ k = 500 km.