Problem 105 - Algebra A Combined Function

Question

The Toroweap Overlook, on the North Rim of the Grand Canyon, lies 3000 vertical feet above the Colorado River. The view is spectacular, and the sheer drop is dramatic. A film crew creating a documentary about the Grand Canyon has suspended a camera platform 296 feet below the Overlook. A camera filter comes loose and falls to the river below. The height of the filter above the river after \(t\) seconds is given by the expression \(2704-16 t^{2}\). a. Find the height of the filter above the river after 3 seconds. b. Find the height of the filter above the river after 7 seconds. c. To the nearest whole second, estimate when the filter lands in the river. d. Factor \(2704-16 t^{2}\).

Step-by-Step Solution

Answer

a: 2560 feet; b: 1920 feet; c: 13 seconds; d: \((52 - 4t)(52 + 4t)\).

Step 1: Identify the Given Expression

The height of the filter above the river after \( t \) seconds is given by the expression \( 2704 - 16t^2 \). This expression will be used to find the height at specific times \( t \).

Step 2: Calculate Height After 3 Seconds

Substitute \( t = 3 \) into the expression: \( 2704 - 16(3)^2 \). Compute the square: \( 3^2 = 9 \). The expression becomes \( 2704 - 16 \times 9 \). Simplify by multiplying: \( 16 \times 9 = 144 \). Subtract: \( 2704 - 144 = 2560 \). The height after 3 seconds is 2560 feet.

Step 3: Calculate Height After 7 Seconds

Substitute \( t = 7 \) into the expression: \( 2704 - 16(7)^2 \). Compute the square: \( 7^2 = 49 \). The expression becomes \( 2704 - 16 \times 49 \). Simplify by multiplying: \( 16 \times 49 = 784 \). Subtract: \( 2704 - 784 = 1920 \). The height after 7 seconds is 1920 feet.

Step 4: Solve for When the Filter Lands in the River

To find out when the filter lands in the river, solve \( 2704 - 16t^2 = 0 \). Rearrange to find \( t^2 \): \( 16t^2 = 2704 \). Divide both sides by 16: \( t^2 = 169 \). Take the square root of both sides: \( t = 13 \). The filter lands in the river after 13 seconds.

Step 5: Factor the Expression

Factor \( 2704 - 16t^2 \) as a difference of squares: \( 2704 = 52^2 \) and \( 16t^2 = (4t)^2 \). Thus, \( 2704 - 16t^2 \) factors to \( (52 - 4t)(52 + 4t) \).

Key Concepts

Factoring Quadratics

Factoring quadratics is a crucial skill in algebra, often used to simplify expressions or solve equations. In the context of this exercise, we need to factor the quadratic expression given by \(2704 - 16t^2\). This expression is known as a "difference of squares." Understanding the concept behind factoring such expressions can make this task easier.

• A difference of squares is an expression in the form \( a^2 - b^2 \).
• It can always be factored into \((a - b)(a + b)\).

For the expression \(2704 - 16t^2\):

• Recognize that \(2704\) is \(52^2\).
• Recognize that \(16t^2\) is \((4t)^2\).

The expression becomes \((52 - 4t)(52 + 4t)\). This method simplifies solving quadratics and recognizing patterns in equations.

Projectile Motion

Projectile motion is a form of motion experienced by an object or particle that is projected near Earth's surface and moves along a curved path under the action of gravity only. In this context, the falling camera filter is a good example of projectile motion, even though it falls vertically. Although equations for projectile motion can get complex, focusing on vertical motion alone simplifies things.

• An object in ideal projectile motion follows a parabolic trajectory.
• Gravity is the only force acting on the object.

The filter's height can be described using a quadratic equation, symbolizing how gravity affects its downward path. By using the equation \(2704 - 16t^2\), we can predict its height at various times, providing insights into its motion until it hits the river.

Solving Quadratic Equations

Solving quadratic equations involves finding the value of \(t\) when the height becomes zero (when the filter hits the river). Quadratic equations often appear in the form \(ax^2 + bx + c = 0\). Here, the exercise uses a simplified version: \(2704 - 16t^2 = 0\).

• First, isolate \(t^2\) by moving terms: \(16t^2 = 2704\).
• Divide both sides by 16 to find \(t^2\): \(t^2 = 169\).
• Take the square root to find \(t\): \(t = 13\) or \(t = -13\).

Only the positive value of \(t\) makes sense in the context of time, giving us the moment when the filter hits the river. Quadratic equations often have two solutions, so interpreting context is important.

Vertical Motion Formula

The vertical motion formula is a key tool in determining the flight path of an object moving under gravity. Typically, it takes the form \(h = h_0 + v_0t - \frac{1}{2}gt^2\), where:

• \(h_0\) is the initial height.
• \(v_0\) is the initial vertical velocity.
• \(g\) is the acceleration due to gravity (approximately 32 ft/s²).

In our exercise, this formula is simplified to \(2704 - 16t^2\) with no initial velocity or additional height. The \(16t^2\) term represents gravitational force acting on the filter, pulling it down as time increases. Understanding the vertical motion formula can help solve similar real-world problems where objects are thrown or dropped.