Question

5. 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.

Answer 1

SOLUTION

GIVEN

A car has six forward gears.

The fastest speed of the car

• 1st gear is 28 km h^-1

• 6th gear is 115 km h^-1

TO DETERMINE

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.

EVALUATION

(a) Arithmetic Sequence :

First term = a = 28

Let Common Difference = d

Now it is given that speed of the car at 6 th gear = 115

$\sf{a + ( 6 - 1) d = 115}$

$\sf{ \implies \: 28 + 5 d = 115}$

$\sf{ \implies \: 5 d = 87}$

$\sf{ \implies \: d = 17.4}$

∴ The fastest speed of the car in 3rd gear

= a + 2d

$\sf{ = 28 + (2 \times 17.4) \: \: km \: {h}^{ - 1} }$

$\sf{ = 28 + 34.8 \: \: km \: {h}^{ - 1} }$

$\sf{ = 62.8 \: \: km \: {h}^{ - 1} }$

(b) Geometric Progression :

First term = a = 28

Let Common Ratio = r

Now it is given that speed of the car at 6 th gear = 115

$\sf{a \times {r}^{5} = 115 }$

$\sf{ \implies \: 28 \times {r}^{5} = 115 }$

$\sf{ \implies \: {r}^{5} = 4.11 }$

$\sf{ \implies \: r = 1.32 }$

∴ The fastest speed of the car in 5th gear

$\sf{ = a \times {r}^{4} }$

$\sf{ = 28 \times {(1.32)}^{4} \: \: \: km \: {h}^{ - 1} }$

$\sf{ = 28 \times 3.04 \: \: \: km \: {h}^{ - 1} }$

$\sf{ = 85 \: \: \: km \: {h}^{ - 1} }$

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