# Rate of change

#### burgerandcheese

##### Junior Member
Q: The height in metres of the water in a harbour is given approximately by the formula h = 6 + 3cos(π/6 * t) where t is the time measured in hours from noon. Find an expression for the rate at which the water is rising at time t. When is it rising fastest?

I know how to to the expression, I got dh/dt = -π/2 sin( π/6 * t)
(By the way, is that an expression or an equation?)

So for the second part of the question when finding the time when it rises the fastest how come we can't set dh/dt = 0 and solve for t? I did it like that and I got my answer wrong

#### HallsofIvy

##### Elite Member
Yes, dh/dt= -π/2 sin((π/6)t). Setting dh/dt= 0 and solving for t tells you when h is a maximum or minimum, not the rate of rising. To find when the water is rising fastest you could differentiate again and set that equal to 0 or, more simply, recognize that, because of the negative, the largest possible value of -π/2 sin((π/6)t) occurs when sin((π/6)t) takes on its smallest value, -1. For what t is sin((π/6)t)= -1?

#### burgerandcheese

##### Junior Member
Right, thanks. That makes sense. Thanks a lot

#### Otis

##### Senior Member
... I got dh/dt = -π/2 sin( π/6 * t)

(By the way, is that an expression or an equation?)
That math statement is an equation because it contains an equals sign.

An equation is a statement that two expressions have the same value.

In the equation above, the expressions are:

dh/dt

-π/2 sin( π/6 * t)

Cheers

#### burgerandcheese

##### Junior Member
That math statement is an equation because it contains an equals sign.

An equation is a statement that two expressions have the same value.

In the equation above, the expressions are:

dh/dt

-π/2 sin( π/6 * t)

Cheers
I see. Alright! Thank you

#### Jomo

##### Elite Member
Otis,
I am wondering if you what you said is always true (An equation is a statement that two expressions have the same value. )

I would call x + 4 = x+5 an equation just because it has an equal sign or would I say it is an invalid equation. I am serious about having doubts after seeing your definition. What do you think?

#### Dr.Peterson

##### Elite Member
Otis,
I am wondering if you what you said is always true (An equation is a statement that two expressions have the same value. )

I would call x + 4 = x+5 an equation just because it has an equal sign or would I say it is an invalid equation. I am serious about having doubts after seeing your definition. What do you think?
It is an equation.

It just happens to be false for all x. A statement that is false is still a statement. An equation that is never true is still an equation.