A scientist is observing a particular species of bacteria growing in a petri dish. The number of bacteria can be represented by the equation n(t)=50(1.2)t

Question

(where n is the number of bacteria after t hours.)

If the scientist checks the population in 4 hours, what would be an appropriate domain for the problem?

Responses

0≤t≤104

0≤n≤4

0≤t≤4

0≤n≤104

Answer 1

In this scenario, the scientist is tracking the growth of bacteria over time. Since the equation \( n(t) = 50(1.2)^t \) describes the number of bacteria as a function of time \( t \) in hours, we need to consider the realistic values that \( t \) can take.

Given that the scientist checks the population after 4 hours, it makes sense to constrain \( t \) to non-negative values from 0 to 4 hours. Therefore, the appropriate domain for \( t \) would be:

0 ≤ t ≤ 4

This option accurately reflects the range of time over which the observation is taking place.

Answer 2

Nora is creating a dog run in her backyard for her dog, Max. She has enough materials to create a run with a perimeter of 120 feet. This is represented by the function A=−w2+60w , where A is the area of the run and w is the width. Which of the following best analyzes the meaning of the x -intercept?(1 point)
Responses

The dog run must have a width between 0 and 60 feet.

The dog run must have a width between 0 and 30 feet.

The dog run must have a width between 0 and 120 feet.

The dog run must have a width between 0 and 900 feet.

Answer 3

To analyze the x-intercept of the function \( A = -w^2 + 60w \), we need to understand what the x-intercepts represent in this context. The x-intercepts occur where the area \( A \) is equal to 0, meaning that the area of the dog run is zero. Since \( A \) is a quadratic function, the x-intercepts will correspond to the width \( w \) values at which the area is zero.

Setting the equation equal to zero for the x-intercepts:

\[ -w^2 + 60w = 0 \]

Factoring out \( w \):

\[ w(-w + 60) = 0 \]

This gives us:

\[ w = 0 \quad \text{or} \quad w = 60 \]

Thus, the x-intercepts are at \( w = 0 \) and \( w = 60 \). The width \( w \) represents the width of the dog run and must therefore be non-negative.

This means that the width of the dog run must be between the point where it has no area (0 feet) and the maximum width that allows for a positive area before reaching 60 feet (where the area again becomes zero).

Consequently, the appropriate response regarding the width of the dog run is:

The dog run must have a width between 0 and 60 feet.

Answer 4

A quadratic function can be used to describe the rate at which an object will fall and predict when it will land or how high it may go.

Using the equation y=−16x2+10, I can predict when an object will return to the ground from being kicked, dropped, or thrown.

What does the 10 in my equation represent?

Responses:

The force of gravity

the maximum height the object will reach

the height of the object when it reaches the ground.

How high the object starts off the ground

Answer 5

In the equation for the quadratic function \( y = -16x^2 + 10 \), the variable \( y \) represents the height of the object at a given time \( x \) (where \( x \) can be thought of as time in seconds after it has been released).

In this equation, the constant term 10 represents the initial height of the object when it is kicked, dropped, or thrown. In other words, when \( x = 0 \) (at the moment of launch), the height \( y \) is equal to 10.

Therefore, the correct interpretation of the 10 in the equation is:

How high the object starts off the ground.