Question

Two sides of a triangle are 6 m and 7 m in length and the angle between them is increasing at a rate of 0.07 rad/s. How do you find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is pi/3?

Answer 1

The overall steps are:

1. Draw a triangle consistent with the given information, labeling relevant information
2. Determine which formulas make sense in the situation (Area of entire triangle based on two fixed-length sides, and trig relationships of right triangles for the variable height)
3. Relate any unknown variables (height) back to the variable `(theta)` which corresponds to the only given rate `((d theta)/(dt))`
4. Do some substitutions into a "main" formula (the area formula) so that you can anticipate using the given rate
5. Differentiate and use the given rate to find the rate you are aiming for `((dA)/(dt))`

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Let's write down the information given formally:

`(d theta)/(dt) = "0.07 rad/s"`

Then you have two fixed-length sides and an angle between them. The third length is a variable value, but it is technically an irrelevant length. What we want is `(dA)/(dt)`. There is no indication that this is a right triangle, however, so let's start by assuming that it's not at the moment.

A theoretically consistent triangle is:

Keep in mind that this is not proportionally representative of the true triangle. The area of this can be found most easily with:

`A = (B*h)/2`

where our base is of course `6`. What is `h`, though? If we draw a dividing line vertically from the apex down to the base, we automatically have a right triangle on the left side of the overall triangle, regardless of the length of side `x`:

Now we do have a right triangle. Notice, however, that our area formula has `h` but not `theta`, and we only know `(d theta)/(dt)`. So, we need to represent `h` in terms of an angle. Knowing that the only known side on the left-hand right triangle is the `7`-lengthed side:

`sintheta = h/7`

`7sintheta = h`

So far, we have:

`(d theta)/(dt) = "0.07 rad/s"` (1)

`A = (Bh)/2` (2)

`7sintheta = color(green)(h)` (3)

So, we can plug (3) into (2), differentiate (2) and implicitly acquire `(d theta)/(dt)`, and plug (1) into (2) to solve for `(dA)/(dt)`, our goal:

`A = (6*color(green)(7sintheta))/2 = 21sintheta`

`color(blue)((dA)/(dt)) = 21costheta((d theta)/(dt))`

`= 21costheta ("0.07 rad/s")`

Finally, at `theta = pi/3`, we have `cos(pi/3) = 1/2` and:

`= 10.5(0.07) = color(blue)("0.735 u"^2"/s")`

(note that `6*7` means the units become `"u"*"u" = "u"^2`, and `2` is not a side length so it had no units. Also, `"rad"` is usually considered to be left out, i.e. `"rad/s" => "1/s"`)