Solved: Rosa joins a club to prepare to run a marathon. During the first training session... [IB Mathematics Applications & Interpretation (Math AI)]

Question

SLPaper 2

Rosa joins a club to prepare to run a marathon. During the first training session Rosa runs a distance of 3000 metres. Each training session she increases the distance she runs by 400 metres.

A marathon is 42.195 kilometres.

In the $k$ th training session Rosa will run further than a marathon for the first time.

Carlos joins the club to lose weight. He runs 7500 metres during the first month. The distance he runs increases by 20% each month.

Write down the distance Rosa runs in the third training session.

[1]

Verified

Solution

3800 m A1

Find the value of $k$ .

[2]

Verified

Solution

$3000 + (k - 1)400 > 42195$ M1

Accept $3 + (k - 1)0.4 > 42.195$

Accept $=$ OR $\geq$

Award M0 for $3000 + (k - 1)400 > 42.195$

$(k =) 99$ A1(ft)G2

Follow through from part (a)(ii), but only if $k$ is a positive integer

Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.

[4]

Verified

Solution

M1 Substitution into sum of an arithmetic series formula:

$\frac{50}{2}(2 \times 3000 + (50 - 1)(400))$

A1 Correct substitution
A1 Calculation result: 640,000 m
A1 Conversion to km: 640 km

Maximum 2 marks if gradient is left as a fraction

Alternative Method

M1 Substitution into sum of an arithmetic series formula (in km):

$\frac{50}{2}(2 \times 3 + (50 - 1)(0.4))$

A1 Correct substitution
A1 Correct conversion of 3000 m and 400 m to km
A1 Final answer: 640 km

Remember to convert all units to km before calculation or convert the final answer to km

Calculate the total distance Carlos runs in the first year.

[3]

Verified

Solution

M1 Substitution into sum of a geometric series formula:

\frac{7500(1.2^{12} - 1)}{1.2 - 1}

A1 Correct substitutions
A1 Final answer: 297,000 m (296,853... m, 297 km)

Maximum 2 marks if r < 1 (distance does not increase) or the final answer is unrealistic (e.g., r = 20)

Remember to express the final answer in meters or kilometers, as appropriate

Find the distance Carlos runs in the fifth month of training.

[3]

Verified

Solution

Method #1

$7500 \times {1.2^{5 - 1}}$ M1A1
$= 15\,600\text{ m }(15\,552\text{ m})$ A1G3

Award (M1) for substitution into geometric series formula, (A1) for correct substitutions.

Method #2

$7.5 \times {1.2^{5 - 1}}$ M1A1
$= 15.6\text{ km}$ A1G3

Award (M1) for substitution into geometric series formula, (A1) for correct substitutions.

Write down the distance Rosa runs in the $n$ th training session.

[2]

Verified

Solution

${3000 + (n - 1)400\text{ m}}$ M1A1

${2600 + 400n\text{ m}}$ M1A1

Award (M1) for substitution into arithmetic sequence formula, (A1) for correct substitution.

Rosa joins a club to prepare to run a marathon. During the first training session Rosa runs a distance of 3000 metres. Each training session she increases the distance she runs by 400 metres.

Alternative Method

Method #1

Method #2

Related topics