Determine the value of a+b if |3a| = 24 cm, |2b| = 10 cm and |3a-2b| = 20
I think something is wrong with your exercise, there is no a and b that match |3a| = 24 cm, |2b| = 10 that can get |3a-2b| = 20 .
Check image.
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This question was given to the class for a test and I have no idea how to do it.
We were given a similar question
imgur(dot)com/a/1P9wGFK
let's suppose a lies along the x-axis, and that 3a-2b makes an angle of θ with a. Then
|-2b|^2 = |3a|^2 + |3a-2b|^2 - 2*|3a|*|3a-2b|*cosθ
100 = 576+400-2*24*20cosθ
θ=24.15°
Then |-2b| makes an angle of 125.1° with |3a|
The components of -2b are thus (-5.75,8.18)
So, b=(2.87,-4.09) and a=(8,0)
and a+b = (10.87,-4.09)
To determine the value of a+b, we need to find the values of a and b separately. Let's start with the equation |3a| = 24 cm.
Since the absolute value of a positive or negative number is equal to its positive value, we can split the equation into two cases:
Case 1: 3a = 24 cm
Dividing both sides of the equation by 3, we get:
a = 8 cm
Case 2: -3a = 24 cm
Dividing both sides of the equation by -3, we get:
a = -8 cm
Next, let's move on to the equation |2b| = 10 cm.
Similar to the previous case, we can split the equation into two cases:
Case 1: 2b = 10 cm
Dividing both sides of the equation by 2, we get:
b = 5 cm
Case 2: -2b = 10 cm
Dividing both sides of the equation by -2, we get:
b = -5 cm
Finally, let's consider the equation |3a - 2b| = 20.
Substituting the values we found for a and b in each case, we have:
Case 1: |3(8) - 2(5)| = 20
Simplifying the equation, we get:
|24 - 10| = 20
|14| = 20
However, absolute value cannot be negative, so this case is not valid.
Case 2: |3(-8) - 2(-5)| = 20
Simplifying the equation gives us:
|-24 + 10| = 20
|-14| = 20
Similarly, absolute value cannot be negative, so this case is also not valid.
Since none of the cases fulfill the equation |3a - 2b| = 20, there are no values of a and b that satisfy all three equations simultaneously. Therefore, there is no solution for a+b in this case.