Word Problems

i. We can use the Revenue formula to determine what the Revenue equation will be.
As we have polynomials for Price (\(2 + 0.10x\)) and Quantity/People (\(240 000 - 10 000x\)), we can expand these using FOIL to get the final equation:

ii. Finding the maximum revenue is similar to finding the vertex for a parabola. In order to do this, we can find the respective roots of the original equations:

Now that we have the roots, we can divide the sum in half to determine the x-coordinate of the vertex. This value represents the # of times the fare should be increased to maximize the revenue:

Therefore, we can determine that the fare should get increased 2 times to maximize the revenue.

iii. To determine the new fare, we need to plug the x-coordinate into the Price equation:

Therefore, we can determine that the new fare that maximizes the revenue is \($2.20\).

iv. To determine the # of riders for the maximum revenue, we need to plug the x-coordinate into the Quantity equation:

Therefore, we can determine that the # of riders needed for maximum revenue is 218 000.

v. We can determine the maximum revenue by plugging the x-coordinate into the Revenue equation:

Therefore, we can determine that the maximum revenue is \($484 000\).

We can start by identifying the equation of the base, based on the description in the question:

We can now plug this new equation into the Area of a Triangle formula:

We can cross-multiply and shift all terms onto one side in order to get a standard quadratic equation:

We can now use the Quadratic Formula to determine the height. We can identify that \(\textcolor{red}{a = 3}\), \(\textcolor{green}{b = 2}\) and \(\textcolor{blue}{c = -616}\):

Using ±, we can determine the height either by adding or subtracting the square root value:

Since the height can't be negative. we can determine that \(h = 14\;[cm]\). We can plug this value into the Base equation to determine the base value:

Therefore, we can determine that \(h = 14\;[cm]\) and \(b = 44\;[cm]\).

First, we can represent each of these values as \(x\), \(x + 1\), \(x + 2\) and \(x + 3\) resepctively. We can then place these values into a single equation:

We can expand this equation, collect like terms and move all terms onto one side to determine the standard quadratic equation:

We can now use the Quadratic Formula to determine \(x\). We can identify that \(\textcolor{red}{a = 4}\), \(\textcolor{green}{b = 12}\) and \(\textcolor{blue}{c = -616}\):

Using ±, we can determine \(x\) either by adding or subtracting the square root value:

We can determine that \(x = 11\) and \(x = -14\). As a result, the consecutive numbers can be either \(11, 12, 13,\) and \(14\) OR \(-14, -13, -12,\) and \(-11\).

i. In order to model an equation for the launch of the rocket, we can use the formula for projectile motion and plug all the given values into it:

Therefore, we can determine that the equation used to model this scenario is \(h = -4.9t² + 100t + 15\).

ii. All we need to do to determine the height after \(2\;[s]\) is plug this value into the equation:

Therefore, we can determine that the rocket's height after \(2\;[s]\) is \(195.4\;[m]\).

iii. In order to determine the maximum height, we need the vertex. We can identify the vertex by completing the square:

Therefore, we can determine that the maximum height of the rocket is approximately \(525\;[m]\).

iv. By determining the vertex of the quadratic in the previous question, we can determine that it took \(10.2\;[s]\) for the rocket to reach maximum height.