Part II: The Answers
284
Answers
401–500
435.
4
9
15
3
Take the derivative of both sides with respect to t:
z
x
y
z dz
dt
x dx
dt
y
dy
dt
3
2
2
2
3
2
2
=
−
=
−
( ) ( )

 

 
To find the value of z, enter x = 4 and y = 1 in the original equation:
z
z
z
3
2
2
3
1 3
4 1
15
15
= −
=
=
Then substitute in the given values along with the value of z and solve for dz
dt
:
3
2
2
3 15
24 3 2 1
2
1 3
2
z dz
dt
x dx
dt
y
dy
dt
dz
dt
( )

 

 
( )
( ) ( )
=
−
=
−
( )( ) ( )( ( )
2
4
9
15
3
dz
dt
=
436.
2.49 m
2
/s
To find the area of the triangle, you need the base and the height. If you let the length
of the base equal 8 meters, then one side of the triangle equals 6 meters. Therefore, if
the height equals h, then h
6
= sinθ so that h = 6 sinθ .
Because the area of a triangle is A
bh
= 1
2
, you can write the area as
A
A
=
=
1
2
8 6
24
( ) sin
sin
θ
θ
This equation now involves θ, so you can take the derivative of both sides with respect
to t to get the rate of increase:
dA
dt
d
dt
= (
)
24 cosθ
θ
( )
Substitute in the given values, θ π
= 6
and d
dt
θ = 0 12
. radians/second, and simplify: