Problem 39 - Elementary and Intermediate Algebra

Question

Quadratic Equation Model Problems Dolphins. Refer to the illustration. The height \(h\) in feet reached by a dolphin \(t\) seconds after breaking the surface of the water is given by \(h=-16 t^{2}+32 t .\) How long will it take the dolphin to jump out of the water and touch the trainer's hand? (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

It takes 1 second for the dolphin to touch the trainer's hand.

Step 1: Identify the Equation

First, identify the given quadratic equation that models the dolphin's jump. The equation given is \[ h = -16t^2 + 32t \]where \( h \) is the height in feet, and \( t \) is the time in seconds. We need to determine the time it takes for the dolphin to reach the maximum height, which would be when it touches the trainer's hand at the peak of its jump.

Step 2: Understanding Maximum Height

A quadratic equation of the form \[ ax^2 + bx + c \]can be used to find the vertex, which gives the maximum or minimum value. This maximum or minimum occurs at \[ t = \frac{-b}{2a} \]for the variable \( t \). In this scenario, \( a = -16 \), \( b = 32 \).

Step 3: Calculate the Time for Maximum Height

Substitute \( a = -16 \) and \( b = 32 \) into the vertex formula for \( t \):\[t = \frac{-32}{2(-16)} = \frac{-32}{-32} = 1 \]Thus, it takes 1 second for the dolphin to reach its maximum height, or the point where it touches the trainer's hand.

Key Concepts

Vertex Formula

In the world of quadratic equations, the Vertex Formula is a powerful tool. It helps us find the peak, or the lowest point, of a parabola. This formula is essential when we deal with motions like the leap of a dolphin. Suppose we have a quadratic equation in the form of \( ax^2 + bx + c \). The vertex, representing either the maximum or minimum point, occurs at \( t = \frac{-b}{2a} \).

• "a" is the coefficient of the squared term.
• "b" is the coefficient of the linear term.
• The vertex formula gives the exact point in time where the gesture or motion reaches its peak.

Understanding this formula is crucial when we want to analyze motions described by quadratic equations, especially when you're curious about the peak of that majestic dolphin leap.

Maximum Height

In any quadratic motion, such as the leap of a dolphin, there is a moment when the height reaches its maximum. This is crucial to know, especially when calculating how high and how far the dolphin can jump. Maximum height is reached at the vertex of the quadratic equation. In our dolphin example, we utilized the equation \( h = -16t^2 + 32t \). The vertex formula gives us the time \( t \) at which this maximum height occurs.

• Plugging in our values, \( a = -16 \) and \( b = 32 \), into the vertex formula \[ t = \frac{-b}{2a} \].
• Solving, \( t = \frac{-32}{-32} = 1 \), tells us that the maximum height occurs at 1 second.
• This tells us not just the time, but it's also when the dolphin touches the trainer's hand.

Being able to find this specific point helps in accurately modeling real-world motions!

Dolphins

Dolphins are not just playful creatures; they are fascinating animals with incredible jumping abilities. When a dolphin leaps out of the water, they're often reaching out to touch an object or a target, like a trainer's hand, which is often the point of maximum height in their jump. When analyzing the dolphin's jump from the water, we use quadratic equations to predict their movements. This approach helps trainers plan interactive displays and ensure safety during shows. The peak of their jump, calculated using the vertex formula and highlighted as maximum height, marks the moment they reach out the highest.

• Understanding these principles is critical for trainers when calculating whether a dolphin can reach certain heights.
• It also assists in enhancing the interactions between trainers and dolphins, making the shows both engaging and safe.

This fascinating blend of marine biology and mathematics illustrates the wonders of nature intertwined with the precise world of math.