Question

problem 3
the graph shows the work done by a mechanics pneumatic car lift at two different temperatures.

a. draw a vertical line on the graph that represents the difference between the work function at the lower temperature, ( u(v) = 5 cdot ln v ), and the work function at the higher temperature, ( q(v) = 8 cdot ln v ), at a domain value of ( v = 2 ) where ( v ) is the volume of the liquid - compression chamber of the car lift.

b. without using a calculator, write and simplify a mathematical expression that represents this difference.

Explanation:

Part (a)

Step1: Identify \( v = 2 \) on x - axis

Locate the value \( v = 2 \) on the horizontal (x - axis) of the given graph.

Step2: Draw vertical line

Draw a vertical line passing through \( v = 2 \) on the x - axis. This vertical line will intersect both the curves \( U(v)=5\cdot\ln v \) and \( Q(v) = 8\cdot\ln v \). The vertical segment between the two intersection points represents the difference between the two functions at \( v = 2 \).

Part (b)

Step1: Write the difference expression

The difference between the work function at higher temperature \( Q(v) \) and the work function at lower temperature \( U(v) \) (or we can also find the difference as \( Q(v)-U(v) \) or \( U(v)-Q(v) \), but since \( Q(v) \) is at higher temperature and \( U(v) \) at lower, if we consider the difference of \( Q(v)-U(v) \)) is given by \( D(v)=Q(v)-U(v) \). Substituting \( Q(v) = 8\cdot\ln v \) and \( U(v)=5\cdot\ln v \), we get \( D(v)=(8\cdot\ln v)-(5\cdot\ln v) \).

Step2: Simplify the expression

Using the distributive property of subtraction over multiplication (or the rule for subtracting like terms with logarithms), we can factor out \( \ln v \). So, \( (8\cdot\ln v)-(5\cdot\ln v)=(8 - 5)\cdot\ln v=3\cdot\ln v \). When \( v = 2 \), the expression is \( 3\cdot\ln 2 \), but the simplified general expression for the difference is \( 3\ln v \) (or if we consider \( U(v)-Q(v) \), it would be \( - 3\ln v \), but since we are looking for the difference, \( Q(v)-U(v)=3\ln v \) is a valid simplified form).

Final Answers

• Part (a): A vertical line passing through \( v = 2 \) on the x - axis (the exact drawing is done as per the steps above on the given graph).
• Part (b): The mathematical expression for the difference is \( 8\ln v-5\ln v \) and its simplified form is \( 3\ln v \).

For part (b) answer:

Step1: Define the difference

The difference between the two functions is \( Q(v)-U(v) \) (or \( U(v)-Q(v) \)). Substitute \( Q(v) = 8\ln v \) and \( U(v)=5\ln v \), so the expression is \( 8\ln v - 5\ln v \).

Step2: Simplify the expression

Using the rule \( a\ln x - b\ln x=(a - b)\ln x \), with \( a = 8 \), \( b = 5 \), we get \( (8 - 5)\ln v=3\ln v \).

Answer:

The expression is \( 8\ln v-5\ln v \) and the simplified form is \( 3\ln v \).