Kenzo is driving his car along a road while his friend records the velocity of the car, $v(t)$, in km/h every minute over a 5-minute period - HSC - SSCE Mathematics Advanced - Question 20 - 2020 - Paper 1
Question 20
Kenzo is driving his car along a road while his friend records the velocity of the car, $v(t)$, in km/h every minute over a 5-minute period. The table gives the velo... show full transcript
Worked Solution & Example Answer:Kenzo is driving his car along a road while his friend records the velocity of the car, $v(t)$, in km/h every minute over a 5-minute period - HSC - SSCE Mathematics Advanced - Question 20 - 2020 - Paper 1
Step 1
Use the trapezoidal rule
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Answer
To find the approximate distance travelled by the car, we will apply the trapezoidal rule. According to this rule, the integral can be approximated as:
∫abf(x)dx≈2(b−a)(f(a)+f(b))+i=1∑n−1f(xi).
In our case:
The time interval [0,5/60] (which is 5 minutes).
The velocities at given intervals are:
v(0)=60 km/h,
v(1/60)=55 km/h,
v(2/60)=65 km/h,
v(3/60)=68 km/h,
v(4/60)=70 km/h,
v(5/60)=67 km/h.
Next, we can calculate the distances using these values.
Step 2
Calculate the approximate distance
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Answer
Using the trapezoidal rule, we calculate:
Approximate distance=25/60−0(v(0)+v(5/60))+i=1∑5/60 at the intervalsv(i)×(1/60)\nTotal=25/60[60+67+2(55+65+68+70)]=2(60)5[60+67+2(55+65+68+70)].
Calculating the sum gives:
1205[60+67+2(55+65+68+70)]=120643≈5.4km.
Thus, the approximate distance covered by the car is 5.4 km.
