--- ### Algebraic Problem: Speed and Distance #### Problem Statement A car travels 210 miles in the same time that a truck travels 150 miles. If the car's speed is 24 mph faster than the truck's, find the car's speed and the truck's speed. #### Instructions **a) Set up an algebraic equation (or equations) to solve the problem.** **b) Solve the equation in part a.** #### Solution Steps **Step A: Setting Up the Equation** Let's define the speeds of the car and the truck as follows: - Let \( t \) be the speed of the truck in mph. - The speed of the car is \( t + 24 \) mph. Since both vehicles travel for the same amount of time, we can use the formula: \[ \text{Distance} = \text{Speed} \times \text{Time} \] For the car: \[ 210 = (t + 24) \times \text{Time} \] For the truck: \[ 150 = t \times \text{Time} \] Since the times are equal, we can set up the following equations: \[ \text{Time for car} = \frac{210}{t + 24} \] \[ \text{Time for truck} = \frac{150}{t} \] Therefore, \[ \frac{210}{t + 24} = \frac{150}{t} \] **Step B: Solving the Equation** Let's solve the equation \(\frac{210}{t + 24} = \frac{150}{t}\): 1. Cross-multiply to get rid of the fractions: \[ 210t = 150(t + 24) \] 2. Expand and simplify the equation: \[ 210t = 150t + 3600 \] 3. Subtract \(150t\) from both sides: \[ 60t = 3600 \] 4. Divide both sides by 60: \[ t = 60 \] Thus, the speed of the truck (\( t \)) is 60 mph. The speed of the car is: \[ t + 24 = 60 + 24 = 84 \text{ mph} \] Therefore, the speed of the car is 84 mph, and the speed of the truck is 60 mph. **Evaluating Functions for Indicated Values** The task is to evaluate the function for the given value: **4) \( f(x) = x^2 - 5x + 7 \); find \( f(-3) \)** In the next steps, we will substitute \( x = -3 \) into the function and solve it to find \( f(-3) \). 1. Start by plugging in \( x = -3 \) into the function: \[ f(-3) = (-3)^2 - 5(-3) + 7 \] 2. Evaluate the expressions inside the function: \[ (-3)^2 = 9 \] \[ -5(-3) = 15 \] 3. Substitute these values back into the function: \[ f(-3) = 9 + 15 + 7 \] 4. Sum the results to get the final value: \[ f(-3) = 31 \] So, \[ f(-3) = 31 \] This method can be used to evaluate any function for an indicated value by substituting the given value into the function and simplifying.