Problem 128 - Introductory Algebra for College Students

Question

You are about to take a great picture of fog rolling into San Francisco from the middle of the Golden Gate Bridge, 400 feet above the water. Whoops! You accidently lean too far over the safety rail and drop your camera. The height, in feet, of the camera after \(t\) seconds is modeled by the polynomial \(400-16 t^{2} .\) The factored form of the polynomial is \(16(5+t)(5-t) .\) Describe something about your falling camera that is easier to see from the factored form, \(16(5+t)(5-t),\) than from the form \(400-16 t^{2}\)

Step-by-Step Solution

Answer

The factored form \(16(5+t)(5-t)\) tells us that the camera hits the water after 5 seconds. This information isn't directly visible from the standard form \(400-16t^2\).

Step 1: Analyzing the factored polynomial

The factored form of the polynomial is \(16(5+t)(5-t)\). This represents the product of two linear factors, where each factor becomes 0 at a certain time \(t\).

Step 2: Identifying the zeros of the polynomial

The zeros of the polynomial are the values of \(t\) for which the height of the camera becomes 0. These can be found by setting each factor of the polynomial equal to 0. The factors 5+t and 5-t become 0 at \(t = -5\) and \(t = 5\) respectively. However, negative time doesn't make sense in this context, so we discard \(t = -5\).

Step 3: Interpretation of the factored form

The zero at \(t = 5\) seconds means that the camera hits the water 5 seconds after it was dropped from the bridge. This piece of information is directly visible from the factored form of the polynomial, but not from the standard form \(400-16t^2\). In other words, the factored form makes it easier to identify when the camera hits the water.

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function that can describe a wide array of real-world situations, such as the path of a falling object, like a camera. These functions are typically expressed in the form \[ ax^2 + bx + c \]where - \(a\), \(b\), and \(c\) are constants, and - \(x\) represents the variable.
In our camera drop scenario, the quadratic function is used to model the height of the camera over time:\[ 400 - 16t^2 \]This expression indicates that the height of the camera starts at 400 feet and decreases as time \(t\) increases.
Quadratic functions are graphed as parabolas, a type of symmetric curve that can open upwards or downwards. In this case, since the quadratic term has a negative coefficient \(-16\), the parabola opens downwards, characterizing the accelerated drop of the camera.

Factored Form

Factored form is a specific way of expressing a polynomial, which makes it easier to identify its roots or zeros. For quadratic functions, the factored form looks like:\[ A(x - r_1)(x - r_2) \]where:

• \(A\) is a constant factor.
• \(r_1\) and \(r_2\) are the roots or zeros of the function.

In our example, the factored form of the height function is:\[ 16(5+t)(5-t) \]This helps to quickly see the interaction of the polynomial with the axis, identifying specific points of interest like when the height becomes zero.
Factored form provides direct insights into the behavior of the polynomial by showing the specific \(t\) values where the polynomial equals zero. It simply unveils the points where the camera reaches the water, offering a clear look at critical moments in the problem.

Zero of a Polynomial

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. Finding these zeros tells us crucial points where something noteworthy happens in the scenario we're modeling. In the case of the falling camera, the zero indicates when the camera hits the water.
To determine the zeros, you set each factored term equal to zero:

• The term \(5 + t = 0\) implies \(t = -5\).
• The term \(5 - t = 0\) implies \(t = 5\).

Since negative time does not make practical sense, we disregard \(t = -5\). Thus, \(t = 5\) is the crucial zero we are interested in. This tells us that 5 seconds after dropping, the camera reaches the water.
Understanding zeros allows us to pinpoint exactly when an important event occurs in the function being examined. It gives a snapshot, like the camera hitting the water, which is important for both understanding the problem and finding a solution.