# Maximum depths of the river

• bagpiper
This will give you the time difference, in hours, between the first minimum and "today".In summary, the depth of the water above the river bed on 7 March 2005, Monday can be represented by the equation D = 9 + 3cos[pi(t)/14], where D is in meters and t is in hours after 12 noon. The minimum depth of the river is 6 meters and the maximum depth is 12 meters. To find the day and time when the water first reaches its minimum, we can take the derivative of D with respect to t and set it equal to 0 to find the time points when D is at its extrema. Then, by looking at the second derivative of D

#### bagpiper

D = 9 + 3cos[pi(t)/14]
where D meters is the depth of the water above the river bed at time t hours after 12 noon on 7 March 2005, Monday. This means that the depth of water only depends on time.
a) Write down the minimum and maximum depths of the river
b) Find the day and time when the water first reaches its minimum.

You know that:
$$-1 \leq \cos \alpha \leq 1$$
So when the D is minimum is when $$\cos \frac{\pi t}{14}$$ is minimum and vice versa, when the D is maximum is when $$\cos \frac{\pi t}{14}$$ is maximum.
Can you go from here?
Viet Dao,

Yup. i got there. So minimum is 6m, maximum is 12m. But i don't know how to manipulate it formula to solve the second part.

Why is pi written as a function of time? Isn't pi a constant? I'll assume it's a typographic error or it means pi * t, i.e. multiplication.

1. Let's say you have D = f(t). If you set f'(t) = 0 that'll give you the extrema (minima and maxima) solutions, say tmin and tmax. Then take each of these solutions and look at f''(t) at this point. If f''(t) < 0 then it's a maximum; if f''(t) > 0 then it's a minimum. If f''(t) = 0 then it's an inflection point, I guess.
2. Now having found all your minima, calculate the time difference between the first such minima occurring (assuming there are more than one, if there is only one minimum then just take its time of occurance, tmin) and "today" (7 March 2005): tmin - t7Mar05.

## 1. What is the maximum depth of the river?

The maximum depth of a river can vary greatly depending on the location and type of river. Some rivers may have a maximum depth of only a few feet, while others can reach depths of hundreds of feet.

## 2. How is the maximum depth of a river determined?

The maximum depth of a river is typically measured by using specialized equipment, such as sonar or depth sounders, which can accurately measure the depth of the river at different points. This data is then used to create a depth profile of the river.

## 3. Can the maximum depth of a river change over time?

Yes, the maximum depth of a river can change over time due to various factors such as erosion, sedimentation, and changes in water flow. Natural events such as floods and droughts can also affect the maximum depth of a river.

## 4. Why is the maximum depth of a river important to know?

Knowing the maximum depth of a river is important for various reasons. It can help with navigation and safety for boats and ships, as well as understanding the overall health and condition of the river. It can also be important for activities such as fishing and water sports.

## 5. Are there any dangers associated with the maximum depth of a river?

While rivers can be beautiful and peaceful, there are also potential dangers associated with their maximum depth. Drowning is a risk for those who are not experienced swimmers or do not have proper safety equipment. Additionally, strong currents and underwater hazards can pose a danger to those in the water.