7 the product of two positive numbers is 100 what

7. The product of two positive numbers is 100. What numbers will produce the least
possible sum? Confirm that the sum is in fact a minimum. (8 marks)
2

x =
√
100
x = 10
y =
100
10
y = 10
s = x + y
s = 10 + 10
s = 20
d
2
dx
2
= x +
100
x
f ' ( x )= 1 −
100
x
2
f ' ' ( x )=
200
x
3
f
' '
( 10 ) =
200
( 10 )
3
f
' ' (
10
)
=
1
5
= 0.20
Therefore, the minimum value is 20.

8.
ABC
is a triangle where
A
is
( 1,3 )
,
B
is
(− 2,1 )
, and
(− 1, − 4 )
. Determine whether
ABC
is a right-angle triangle.
C
is
Task #3: Communication Questions
9. Discuss how displacement, velocity, and acceleration are related to a function and its
derivatives. Create an equation for the displacement of an object and determine the
equations for the velocity and acceleration of the object at any time
t
. (6 marks)

t = 0.59 s
 Equation for Velocity:
v ( t ) = s ' ( t )
v ( t ) = 0.12 ( t )
v ( 0.59 ) = 0.12 ( 0.59 )
v ( 0.59 ) = 0.071 m / s
 Equation for Acceleration:
a
(
t
)
= v
' (
t
)
a
(
t
)
= 0.24 m / s
2
10. What is the difference between a scalar and a vector quantity? Include an example of
each. (4 marks)
Scalar Quantity Example: Area
Vector Quantity Example: Acceleration
11. List and describe the three properties of vector arithmetic. Give an example of each
property using vectors in Cartesian form. (12 marks: 4 marks for each property)
)

¿ (− 1,13 )
⃗ u + ( ⃗ v + ⃗ w ) = ( 2,3 ) +
[
( − 1,5 ) + ( − 2,5 )
]
− 1 − 2
¿
( 5 + 5 , ( ¿ )
]
¿
(
2,3
)
+ ¿
¿ ( 2,3 ) + ( − 3,10 )
¿ (− 1,13 )
Therefore,
( ⃗ u + ⃗ v ) + ⃗ w = ⃗ u +( ⃗ v + ⃗ w )
.
2. Distributive Property of Scalar Multiplication
 For any scalar quantity a and vector quantities
⃗ u ∧ ⃗ v
:
a ( ⃗ u + ⃗ v ) = a ⃗ u + a ⃗ v
o Example:
a ( ⃗ u + ⃗ v )
= 5[(4, 3,7) + (-3, 0, 6)]
= [(20, 15, 35) + (-15, 0, 30)]
= (5, 15, 65)
a ⃗ u + a ⃗ v
= 5(4, 3, 7) + 5(-3, 0, 6)
= (20, 15,35) + (-15, 0, 30)
= (5, 15, 65)
Therefore,
a
(
⃗ u + ⃗ v
)
= a ⃗ u + a ⃗ v
.
3. Associative Property of Vector Addition

 For any three vectors
⃗ u, ⃗ v ∧ ⃗ w :
(
⃗ u + ⃗ v
)
+ ⃗ w = ⃗ u +( ⃗ v + ⃗ w )
o Example:
( ⃗ u + ⃗ v ) + ⃗ w =
[
( 3,5 ) + ( − 6,4 )
]
+ ( − 2,0 )
¿
[
( 3 − 6 ) , ( 5 + 4