Evaluate [r]; i.e., the units of r, in the expression
r =v (s - a) (s - b) (s - c)/s where s =.5 (a + b + c) , and a, b, and c have
units of meters.
1. [r] =1/m^1/3
2. [r] = m^1/2
3. [r] = m
4. [r] = m^2
5. [r] = m^1/3
6. [r] =1/m^2
7. [r] = m^3
8. [r] =1/m^1/2
9. [r] =1/m
3.
s=m and m^3/m is m^2.
the square root will be m.
To evaluate the units of r, we need to substitute the units of the variables a, b, and c into the equation for r and simplify.
Given:
r = v (s - a) (s - b) (s - c) / s
And:
s = 0.5(a + b + c)
Substituting the value of s into the expression for r, we get:
r = v (0.5(a + b + c) - a) (0.5(a + b + c) - b) (0.5(a + b + c) - c) / 0.5(a + b + c)
Simplifying the expression, we get:
r = v (0.5a + 0.5b + 0.5c - a) (0.5a + 0.5b + 0.5c - b) (0.5a + 0.5b + 0.5c - c) / 0.5(a + b + c)
r = v (-0.5a + 0.5b + 0.5c) (-0.5a + 0.5b - 0.5c) (0.5a - 0.5b + 0.5c) / 0.5(a + b + c)
r = v (-0.5)(0.5)(-0.5a + 0.5b + 0.5c)(-0.5a + 0.5b - 0.5c)(0.5a - 0.5b + 0.5c) / (0.5)(a + b + c)
Cancelling out the common factors, we get:
r = (-0.125)(-a + b + c)(-a + b - c)(a - b + c) / (a + b + c)
Now let's analyze the units:
- The velocity v is typically measured in m/s, but since we're only evaluating the units of r, we don't need to consider the units of v.
- The terms (-a + b + c), (-a + b - c), and (a - b + c) all have units of meters since a, b, and c have units of meters.
- The denominator (a + b + c) also has units of meters since a, b, and c have units of meters.
We can see that the units of r are the product of the units of the three terms in the numerator divided by the units of the denominator, which results in meters cubed: [r] = m^3.
Therefore, the correct answer is option 7: [r] = m^3.