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# Step-by-Step Math Answer

Substitute
*
π
*
/2 for
*
t
*
in
*
x
*
_{
1
}
and simplify.

To find the rate at which the distance between the two objects changes, first find
*
x
*
_{
1
}
.

So, substitute
*
π
*
/2 for
*
t
*
and simplify.

*
x
*
_{
1
}
= 10 cos 2(
*
π
*
/2) or –10

Repeat the above procedure to evaluate
*
x
*
_{
2
}
,
*
y
*
_{
1
}
, and
*
y
*
_{
2
}
.

Next, repeat the above procedure to evaluate
*
x
*
_{
2
}
,
*
y
*
_{
1
}
, and
*
y
*
_{
2
}
.

*
x
*
_{
2
}
= 0

*
y
*
_{
1
}
= 0

*
*
and

*
y
*
_{
2
}
= 4

Use the Distance Formula.

The distance
*
s
*
between the two object is

.

Substitute the known values and simplify.

Evaluate the partial derivative of
*
s
*
with respect to
*
x
*
_{
1
}
when
*
t
*
=
*
π
*
/2.

Now, evaluate the partial derivative of
*
s
*
with respect to
*
x
*
_{
1
}
.

Substitute the known values and simplify.

Find the remaining partial derivatives of
*
s
*
when
*
t
*
=
*
π
*
/2.

Repeat the above procedure to find the remaining partial derivatives of
*
s
*
when
*
t
*
=
*
π
*
/2.

Find the derivative of
*
x
*
_{
1
}
when
*
t
*
=
*
π
*
/2.

To find the derivative of
*
x
*
_{
1
}
when
*
t
*
=
*
π
*
/2, first differentiate
*
x
*
_{
1
}
with respect to
*
t
*
.

Next, substitute
*
π
*
/2 for
*
t
*
and simplify.

Find the derivatives of
*
x
*
_{
2
}
,
*
y
*
_{
1
}
, and
*
y
*
_{
2
}
when
*
t
*
=
*
π
*
/2.

Repeat the above procedure to find the derivatives of
*
x
*
_{
2
}
,
*
y
*
_{
1
}
, and
*
y
*
_{
2
}
when
*
t
*
=
*
π
*
/2.

Use the Chain Rule for one independent variable.

Let
*
w
*
=
*
f
*
(
*
x
*
,
*
y
*
), where
*
f
*
is a differentiable function of
*
x
*
and
*
y
*
. If
*
x
*
=
*
g
*
(
*
t
*
) and
*
y
*
=
*
h
*
(
*
t
*
), where
*
g
*
and
*
h
*
are differentiable functions of
*
t
*
, then
*
w
*
is a differentiable function of
*
t
*
, and

.

So,

.

Substitute the known values.

Simplify.

Simplify.

So, the distance between the two objects is changing at a rate of –2.04.