A company, which is making 200 mobile phones each week, plans to increase its production. The number of mobile phones produced is to be increased by 20 each week from 200 in week 1 to 220 in week 2, to 240 in week 3 and so on, until it is producing 600 in week N. (a) Find the value of N. The company then plans to continue to make 600 mobile phones each week. (b) Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week 1.

A-Level Edexcel Maths Pure Differentiation: A company, which is making 200 m

A company, which is making 200 mobile phones each week, plans to increase its production - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 1

Question 9

A company, which is making 200 mobile phones each week, plans to increase its production. The number of mobile phones produced is to be increased by 20 each week fr... show full transcript

Worked Solution & Example Answer:A company, which is making 200 mobile phones each week, plans to increase its production - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 1

Step 1

Find the value of N.

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Answer

To find the value of N, we first identify the pattern in the production of mobile phones:

Week 1: 200
Week 2: 220
Week 3: 240
...

The production increases by 20 each week. We can express this as:

$P(n) = 200 + 20(n-1)$

We need to find when the production reaches 600:

$600 = 200 + 20(N-1)$

Simplifying this equation:

$600 - 200 = 20(N - 1)$ $400 = 20(N - 1)$ $N - 1 = rac{400}{20}$ $N - 1 = 20$ $N = 21$

Therefore, the value of N is 21.

Step 2

Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week 1.

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Answer

First, we recognize that the production in the first 21 weeks follows an arithmetic progression (AP), with:

First term, $a = 200$
Common difference, $d = 20$
Number of terms, $n = 21$

The sum of the first n terms of an AP is given by:

$S_n = \frac{n}{2} \cdot (2a + (n - 1)d)$

Substituting the known values: $S_{21} = \frac{21}{2} \cdot (2 \cdot 200 + (21 - 1) \cdot 20)$ $= \frac{21}{2} \cdot (400 + 400)$ $= \frac{21}{2} \cdot 800$ $= 21 \cdot 400 = 8400$

In weeks 22 to 52, the company produces a constant 600 mobile phones per week. There are 31 weeks in this range:

$ext{Total from weeks 22 to 52} = 600 \cdot 31 = 18600$

Thus, the total number of mobile phones produced in the first 52 weeks:

$ext{Total} = S_{21} + ext{Total from weeks } 22 ext{ to } 52$

$= 8400 + 18600 = 27000$

Therefore, the total number of mobile phones made in the first 52 weeks is 27,000.

A company, which is making 200 mobile phones each week, plans to increase its production - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 1

Worked Solution & Example Answer:A company, which is making 200 mobile phones each week, plans to increase its production - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 1

Find the value of N.

Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week 1.

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