There are several vehicles in a parking lot. Some of them are motorcycles (with 2 wheels), and some are cars (with 4 wheels). There are 10 vehicles in
the lot, and there are 32 wheels. How many vehicles of each type are in the
lot?

There are four motorcycles and six cars.

Step-by-step explanation:

Let m represent the number of motorcycles and c represent the number of cars.

Since there are ten vehicles in total, the sum of the number of motorcycles and the number of cars must total ten. Hence:

And since each motorcycle has two wheels and each car has four wheels and there are 32 wheels in total:

Solve the system of equations. First, we can divide the second equation by two:

From the first equation, we can subtract c from both sides:

Substitute:

Simplify:

Therefore:

There are six cars.

Using the modified equation:

Solve for m:

So, there are four motorcycles and six cars.

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.