Where do they meet? Kelly started at noon (t=0) riding a bike from Niwot to Berthoud, a distance of 20 km, with velocity v(t) = 15 / (t + 1)² (decreasing because of fatigue). Sandy started at noon (t=0) riding a bike in the opposite direction from Berthoud to Niwot with velocity u(t) = 20 / (t + 1)² (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours.

c. When do they meet? How far has each person traveled when they meet?

검증된 단계별 안내

Define the position functions for Kelly and Sandy based on their velocities and starting points. Since Kelly starts at Niwot (position 0) and travels toward Berthoud (position 20 km), her position at time \(t\) is given by integrating her velocity \(v(t) = \frac{15}{(t+1)^2}\) from 0 to \(t\): \[ K(t) = \int_0^t \frac{15}{(s+1)^2} \, ds \]

Similarly, Sandy starts at Berthoud (position 20 km) and travels toward Niwot (position 0 km) with velocity \(u(t) = \frac{20}{(t+1)^2}\). Since she moves in the opposite direction, her position at time \(t\) is: \[ S(t) = 20 - \int_0^t \frac{20}{(s+1)^2} \, ds \]

Calculate the integrals for both \(K(t)\) and \(S(t)\) to express their positions explicitly as functions of \(t\). Recall that \[ \int \frac{1}{(s+1)^2} ds = -\frac{1}{s+1} + C \]

Set the positions equal to find the meeting time \(t_m\): \[ K(t_m) = S(t_m) \] This equation means they are at the same point on the 20 km path at time \(t_m\).

Solve the equation from step 4 for \(t_m\). Once \(t_m\) is found, substitute it back into \(K(t)\) and \(S(t)\) to find how far each person has traveled when they meet.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

11m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Position as an Integral of Velocity

To find the position of each cyclist over time, we integrate their velocity functions with respect to time. Since velocity is the rate of change of position, integrating v(t) or u(t) from 0 to t gives the distance traveled by each cyclist at time t.

Relative Motion and Meeting Point

Kelly and Sandy start from opposite points and move toward each other. They meet when the sum of the distances each has traveled equals the total distance between Niwot and Berthoud (20 km). Setting the sum of their positions equal to 20 allows us to solve for the meeting time.

Solving Equations Involving Definite Integrals

Determining when they meet requires solving an equation involving definite integrals of their velocity functions. This involves calculating integrals, setting up an equation for total distance, and solving for the time variable t, often requiring algebraic manipulation or numerical methods.