A car has six forward gears. The fastest speed of the car - in 1st gear is 28 km h⁻¹ - in 6th gear is 115 km h⁻¹ Given that the fastest speed of the car in successive gears is modelled by an arithmetic sequence. (a) find the fastest speed of the car in 3rd gear. Given that the fastest speed of the car in successive gears is modelled by a geometric sequence, (b) find the fastest speed of the car in 5th gear.

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A car has six forward gears - Edexcel - A-Level Maths Pure - Question 7 - 2020 - Paper 1

Question 7

A car has six forward gears. The fastest speed of the car - in 1st gear is 28 km h⁻¹ - in 6th gear is 115 km h⁻¹ Given that the fastest speed of the car in succes... show full transcript

Worked Solution & Example Answer:A car has six forward gears - Edexcel - A-Level Maths Pure - Question 7 - 2020 - Paper 1

Step 1

(a) find the fastest speed of the car in 3rd gear.

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Answer

To find the speed in the 3rd gear using an arithmetic sequence:

We know the speed in the 1st gear ( a = 28 , \text{km h}^{-1} ) and in the 6th gear ( a + 5d = 115 , \text{km h}^{-1}), where d is the common difference.
Using the equation from the 6th gear: $115 = 28 + 5d$ Solving for d: $5d = 115 - 28$ $5d = 87$ $d = \frac{87}{5} = 17.4 \, \text{km h}^{-1}$
Now, substitute d back to find the speed in the 3rd gear: $\text{Speed in 3rd gear} = a + 2d$ $= 28 + 2(17.4)$ $= 28 + 34.8$ $= 62.8 \, \text{km h}^{-1}$

Step 2

(b) find the fastest speed of the car in 5th gear.

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Answer

To find the speed in the 5th gear using a geometric sequence:

Based on the given speeds, we understand that these follow a geometric progression, hence: $\text{Let } r \text{ be the common ratio.}$

From the 1st and 6th gear: $115 = 28 r^5$
Rearranging for r: $r^5 = \frac{115}{28}$ $r = \left(\frac{115}{28}\right)^{1/5} \approx 1.3265$
Using this value of r, we can find the speed in the 5th gear: $\text{Speed in 5th gear} = 28 r^4$ $= 28 \times (1.3265)^4$ $\approx 86.7 \, \text{km h}^{-1}$

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