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# Step-by-Step Math Answer

Give the formula for the area
*
A
*
of the triangle.

The area of the triangle is

*
A
*
= (1/2)
*
bh
*
,

where
*
b
*
is the base (which we will take to be the side
**
opposite
**
angle
*
θ
*
) and
*
h
*
is the height of the triangle.

Express
*
b
*
and
*
h
*
as functions of the angle
*
θ
*
and the length
*
x
*
of a leg of the triangle.

We express
*
b
*
and
*
h
*
as functions of the angle
*
θ
*
and the length
*
x
*
of a leg of the triangle:

*
b
*
/2 =
*
x
*
[sin(
*
θ
*
/2)].

Multiplying both sides of this equation by 2, we get

*
b
*
= 2
*
x
*
[sin(
*
θ
*
/2)].

Also,

*
h
*
=
*
x
*
[cos(
*
θ
*
/2)].

Substitute these results into the expression for
*
A
*
.

Substituting these results, we have

Apply the double–angle identity for sin(
*
θ
*
).

We apply the double–angle identity for sin(
*
θ
*
):

sin
*
θ
*
= 2[sin(
*
θ
*
/2)][cos(
*
θ
*
/2)].

Dividing through by 2, we get

(1/2)(sin
*
θ
*
) = [sin(
*
θ
*
/2)][cos(
*
θ
*
/2)],

so

Find ∂
*
A
*
/ ∂
*
x
*
and ∂
*
A
*
/ ∂
*
θ
*
.

We find ∂
*
A
*
/ ∂
*
x
*
and ∂
*
A
*
/ ∂
*
θ
*
:

What independent variable(s) do
*
x
*
and
*
θ
*
depend on?

They depend on time,
*
t
*
.

*
x
*
depends on
*
θ
*
, and
*
θ
*
depends on
*
x
*
.

*
x
*
and
*
θ
*
are functions of a single variable, time, which we denote here by
*
t
*
.

Give an expression for
*
dA
*
/
*
dt
*
, the rate at which
*
A
*
is changing.

The rate at which
*
A
*
is changing is

Substitute the given data.

Substituting the given data, we obtain

in units of square meters per hour.