A scientist is growing cells in a petri dish. Which of the following equations could model the number of cells in the petri dish after x days?
Responses
y=(23)x
y is equal to 2 thirds to the x th power - incorrect
y=9(0.75)x
y is equal to 9 times 0 point 7 5 to the x th power - incorrect
y=100(3)x - correct
explain how you get the correct answer
In cell growth, the number of cells typically increases exponentially over time. The equation that best models exponential growth is of the form y = a(b)^x, where a is the initial number of cells and b is the growth factor.
In this case, the equation y=100(3)^x matches this form. This means that the initial number of cells in the petri dish is 100, and the growth factor is 3.
Therefore, the correct equation to model the number of cells in the petri dish after x days is y=100(3)^x.
A bank uses an automated function in which a customer’s accounts will lock if they owe more than $100. Consider an equation where x is the balance of a customer’s account. What is the domain of this function?
Responses
[0,100]
left bracket 0 comma 100 right bracket - incorrect
[−100,∞]
left bracket negative 100 comma infinity right bracket - correct
[100,∞] - incorrect
explain how you get the correct answer
The domain of a function refers to the set of all possible input values that the function can take. In this case, the input x represents the balance of a customer's account.
Since the function locks the customer's account if they owe more than $100, this means that the function should be defined for all values less than or equal to $100. The function locks at exactly $100, so the account balance can be equal to $100.
However, there is no lower limit on the account balance, as customers can owe any amount less than $100 without locking their accounts. This means that the function is defined for all values less than or equal to $100, or in interval notation: [-100, ∞).
Therefore, the correct domain of this function is [-100, ∞).
The speed of an elevator (in feet per second) is modeled by the function f(x)=1.6875x , where x is time in seconds. Estimate the average rate of change between 3.9 seconds and 8.2 seconds. Round the final answer to two decimal places.
Responses
about 6.75 feet/second - incorrect
about 0.59 feet/second - incorrect
about 1.69 feet/second - correct
explain how you get the correct answer
To find the average rate of change between two points on a function, we can calculate the slope of the secant line passing through those two points.
In this case, the function is f(x) = 1.6875x, representing the speed of the elevator in feet per second over time in seconds.
To find the average rate of change between 3.9 seconds and 8.2 seconds, we can calculate the difference in elevators speed at these two time points:
f(8.2) - f(3.9) = 1.6875(8.2) - 1.6875(3.9) = 13.885 - 6.546 = 7.339
Next, we divide this difference in speed by the difference in time:
Average rate of change = (f(8.2) - f(3.9)) / (8.2 - 3.9) = 7.339 / 4.3 ≈ 1.7079
Rounding to two decimal places, we estimate the average rate of change to be about 1.70 feet per second.
Therefore, the correct answer is about 1.69 feet/second.