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Lincoln’s parents invested $5,000 in a college savings account when he was born. If the account has a growth rate of 12%, the situation can be modeled by the equation a(t)=5,000(1.12)t , where a is the amount after t years. Which one of the following statements is true about the graph of the function? (1 point) Responses As t increases, a increases slowly at first and then quickly. As t increases, a increases slowly at first and then quickly. As t increases, a decreases quickly at first and then slowly. As t increases, a decreases quickly at first and then slowly. Skip to navigation

5 answers

The given function \( a(t) = 5,000(1.12)^t \) represents an exponential growth model. In this equation:

  • \( 5,000 \) is the initial amount (the principal).
  • \( 1.12 \) is the growth factor, which means the account grows by \( 12% \) each year.
  • \( t \) is the number of years.

As \( t \) increases, the value of \( a(t) \) will increase because of the compounding effect of the growth rate.

For exponential functions like this one, the growth starts off relatively slow, particularly for small values of \( t \), but as time goes on, the amount increases more rapidly due to the nature of exponential growth. This means that the graph of \( a(t) \) rises slowly at first and then rises quickly as \( t \) increases.

Thus, the correct statement is:

As t increases, a increases slowly at first and then quickly.

Question 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 30 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 120 feet. The dog run must have a width between 0 and 900 feet. The dog run must have a width between 0 and 900 feet. The dog run must have a width between 0 and 60 feet. The dog run must have a width between 0 and 60 feet.

The function given is \( A = -w^2 + 60w \), which represents the area \( A \) of the dog run as a function of width \( w \). To analyze the meaning of the \( x \)-intercept of this function, we need to find the values of \( w \) at which the area \( A \) is equal to zero.

Setting the equation equal to zero:

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

Factoring the equation:

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

This gives us two solutions:

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

The \( x \)-intercept occurs at these two values.

  1. \( w = 0 \) means that if the width of the dog run is zero, the area is zero, which is not a practical width for a dog run.
  2. \( w = 60 \) indicates the maximum width the dog run can have while still allowing for a non-zero area.

Given the context of the problem and the constraints represented by the perimeter (the width must also lead to a valid area), the width of the dog run must be confined between these two intercepts:

  • Thus, the viable width \( w \) must be in the range \( 0 < w < 60 \) feet.

Therefore, the correct analysis of the \( x \)-intercept is:

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

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? (1 point) Responses The force of gravity The force of gravity the height of the object when it reaches the ground. the height of the object when it reaches the ground. the maximum height the object will reach the maximum height the object will reach How high the object starts off the ground

In the quadratic equation \( y = -16x^2 + 10 \), where \( y \) represents the height of the object and \( x \) represents time, the value \( 10 \) is the constant term.

This constant term represents the initial height from which the object is thrown, kicked, or dropped at \( x = 0 \) (when time starts). Therefore, it indicates how high the object starts off the ground.

So, the correct interpretation of the \( 10 \) in this equation is:

How high the object starts off the ground.