Modelling with polynomials

[anzgurl]

5 A particle moves horizontally in a straight line according to the rule
x(t) = t^3 − 5t^2 + 3t − 5, 0 ≤ t ≤ 7
where x metres is its displacement to the right of the origin at time t seconds.
a What is the initial position of the particle?
b After how many seconds does the particle pass through the origin, correct to 2 decimal
places?

i need help on part b. could some one help me please?
thank you

 

[danago]

When it passes the origin, the displacement is 0 right? Therefore, you could say that x(t)=0 when it passes the origin. So just find the root of the equation that suits the given domain.

 

[Architeuthis Dux]

for your question. Let me provide a response to your query about the particle's motion.

a) The initial position of the particle can be found by plugging in t=0 into the given equation:
x(0) = 0^3 - 5(0)^2 + 3(0) - 5 = -5
Therefore, the initial position of the particle is -5 metres to the right of the origin.

b) To find the time at which the particle passes through the origin, we need to set x(t) = 0 and solve for t:
0 = t^3 - 5t^2 + 3t - 5
We can use a graphing calculator or a polynomial solver to find the roots of this equation. The roots are t ≈ 1.28, t ≈ 3.16, and t ≈ 4.56. However, we are only concerned with the time when the particle passes through the origin, which is t ≈ 1.28 seconds (since it is the only positive root).

Therefore, after 1.28 seconds, the particle will pass through the origin. We can round this to 2 decimal places to get the final answer of 1.28 seconds.

I hope this helps. Let me know if you have any further questions.