As a crude approximation, we assume that the (constant) speed at which the birds make theirjourney increases linearly with the time they spend foraging for energy before commencing it,through the following formula:v0 = 50 + 3 τ, (8)where τ is the time in days that the red knot spends foraging in the Dutch Wadden Sea and v0 isthe constant speed in km per day at which it will be able to travel the entire journey of 9000 km.(b). Explain that the total time before the red knot arrives in South Africa will be given byT = 900050 + 3 τ + τ.
Question
As a crude approximation, we assume that the (constant) speed at which the birds make theirjourney increases linearly with the time they spend foraging for energy before commencing it,through the following formula:v0 = 50 + 3 τ, (8)where τ is the time in days that the red knot spends foraging in the Dutch Wadden Sea and v0 isthe constant speed in km per day at which it will be able to travel the entire journey of 9000 km.(b). Explain that the total time before the red knot arrives in South Africa will be given byT = 900050 + 3 τ + τ.
Solution
The formula given is v0 = 50 + 3τ, where τ is the time spent foraging and v0 is the constant speed of travel. This means that the speed of the bird increases linearly with the time spent foraging.
The total distance the bird needs to travel is 9000 km. The time it takes to travel this distance at a constant speed is given by the formula time = distance/speed. Substituting the given formula for speed, we get time = 9000/(50 + 3τ).
However, this is only the time spent travelling. The bird also spends τ days foraging before it starts its journey. Therefore, the total time before the bird arrives in South Africa is given by the time spent foraging plus the time spent travelling, which is T = τ + 9000/(50 + 3τ).
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