Questions: A stepladder is diagrammed on a coordinate plane. The base of one leg of the stepladder is at the point ( 0,0 ), the base of the other leg is 4 feet to the right, and the top of the stepladder is 5.5 feet high. a. Write a rule for function f to represent the stepladder, and state the domain. b. Another stepladder has legs 6 feet apart and is 8.25 feet high. The base of its left leg is at the point ( 0,0 ). Write a rule for function g to represent this ladder and describe it as a transformation of t. a. f(x)= (Simplify your answer. Use integers or decimals for any numbers in the expression.)

A stepladder is diagrammed on a coordinate plane. The base of one leg of the stepladder is at the point ( 0,0 ), the base of the other leg is 4 feet to the right, and the top of the stepladder is 5.5 feet high. a. Write a rule for function f to represent the stepladder, and state the domain. b. Another stepladder has legs 6 feet apart and is 8.25 feet high. The base of its left leg is at the point ( 0,0 ). Write a rule for function g to represent this ladder and describe it as a transformation of t. a. f(x)= (Simplify your answer. Use integers or decimals for any numbers in the expression.)
Transcript text: A stepladder is diagrammed on a coordinate plane. The base of one leg of the stepladder is at the point ( 0,0 ), the base of the other leg is 4 feet to the right, and the top of the stepladder is 5.5 feet high. a. Write a rule for function f to represent the stepladder, and state the domain. b. Another stepladder has legs 6 feet apart and is 8.25 feet hight. The base of its left leg is at the point ( 0,0 ). Write a rule for function g to represent this ladder and describe it as a transformation of $t$. a. $f(x)=\square$ $\square$ (Simplify your answer. Use integers or decimals for any numbers in the expression.)
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Solution

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Solution Steps

Solution Approach
  1. For part (a), we need to find the equation of the line representing the stepladder. The line passes through the points (0,0) and (4,5.5). We can use the slope-intercept form of a line, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
  2. For part (b), we need to find the equation of the line for the second stepladder, which passes through the points (0,0) and (6,8.25). We can again use the slope-intercept form.
  3. To describe the transformation, we compare the slopes and intercepts of the two lines.
Step 1: Equation of the First Stepladder

The first stepladder has its base at the point \((0,0)\) and the top at \((4, 5.5)\). The slope \(m_1\) is calculated as follows:

\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5.5 - 0}{4 - 0} = 1.375 \]

Since the line passes through the origin, the y-intercept \(b_1\) is \(0\). Therefore, the equation of the first stepladder is:

\[ f(x) = 1.375x + 0 \]

The domain of this function is:

\[ \text{Domain of } f: [0, 4] \]

Step 2: Equation of the Second Stepladder

The second stepladder has its base at \((0,0)\) and the top at \((6, 8.25)\). The slope \(m_2\) is calculated as follows:

\[ m_2 = \frac{y_4 - y_3}{x_4 - x_3} = \frac{8.25 - 0}{6 - 0} = 1.375 \]

Again, since the line passes through the origin, the y-intercept \(b_2\) is \(0\). Therefore, the equation of the second stepladder is:

\[ g(x) = 1.375x + 0 \]

The domain of this function is:

\[ \text{Domain of } g: [0, 6] \]

Step 3: Transformation Description

The slopes of both stepladders are equal, \(m_1 = m_2 = 1.375\). This indicates that the second stepladder is a scaled version of the first, with a slope ratio of:

\[ \text{Slope Ratio} = \frac{m_2}{m_1} = 1.0 \]

Final Answer

The equations and domains of the stepladders are as follows:

  • For the first stepladder: \(f(x) = 1.375x\) with domain \([0, 4]\)
  • For the second stepladder: \(g(x) = 1.375x\) with domain \([0, 6]\)

The transformation indicates that the second stepladder is a scaled version of the first with a slope ratio of \(1.0\).

\[ \boxed{f(x) = 1.375x, \quad g(x) = 1.375x} \]

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