Let x be the length of a fire truck and y the length of an ambulance. How does the 1 foot of space in between vehicles affect the equation?
130 feet
Practice makes perfect
First, we need to find the lengths of each type of vehicle. Then, we can find the necessary length of the parking lane for 1 fire truck and 5 ambulances.
Finding the Vehicle Lengths
We will let x represent the length of the fire truck and y be the length of an ambulance. Let's label the given figure to create our equations. We keep in mind that the distance between each vehicle is 1 foot.
By looking at the figure we can see that 3 fire trucks, 4 ambulances, and 6 one foot long spaces can fit into a 152 foot parking lane. Let's write an equation describing the situation.
3 x+4 y+6( 1)=152We also know that 2 fire trucks, 5 ambulances, and 6 one foot spaces can fit into a 136 foot parking lane. Let's write this as an equation.
2 x+5 y+6( 1)=136
By combining these we get a system of equations. Since we are only interested in the lengths of the vehicles, we can simplify the equations.
3 x+4 y+6( 1)=152 2 x+5 y+6( 1)=136 ⇒ 3x+4y=146 2x+5y=130
In order to solve this using the Elimination Method we will need to multiply the equations to have opposite coefficients for one of the variables. If we multiply the first equation by -2 and the second equation by 3, the x-variables will be opposites.
3x+4y=146 2x+5y=130 ⇒ -6x-8y=-292 6x+15y=-390
We can now solve the system of equations.
We have found that the length of a fire truck is 55 feet and the length of an ambulance is 14 feet.
Finding the Length of the Parking Lane
We need to know how long a parking lane needs to be for one fire truck and five ambulances. Let's illustrate this with an image.
We see that we need to add 5 one foot spaces to the expression when we describe this situation.
l = x + 5 y +5( 1)
When we substitute the values found above into this expression, we get the total length l of the parking lane occupied by vehicles.
