What is the range of G(t)?
Understand the Problem
The question is asking for the range of the function G(t), which represents the number of gigabytes downloaded in t minutes. Given that the download speed is 0.5 gigabytes per minute and the total download is 35 gigabytes over 70 minutes, we need to determine the possible values of G(t) for t ranging from 0 to 70 minutes.
Answer
The range of $G(t)$ is all real numbers from $0$ to $35$.
Answer for screen readers
The range of $G(t)$ is all real numbers from $0$ to $35$.
Steps to Solve
- Define the Function G(t)
The function $G(t)$ represents the number of gigabytes downloaded at a rate of 0.5 gigabytes per minute. Thus, we can express it as: $$ G(t) = 0.5t $$
- Determine the Range for t
Since $t$ represents time in minutes and Nolan can download for a maximum of 70 minutes, the values of $t$ will range from 0 to 70: $$ 0 \leq t \leq 70 $$
- Calculate the Corresponding Values for G(t)
Now, we find the minimum and maximum values of $G(t)$ by substituting the limits of $t$:
- For $t = 0$: $$ G(0) = 0.5 \times 0 = 0 $$
- For $t = 70$: $$ G(70) = 0.5 \times 70 = 35 $$
- Identify the Range of G(t)
From the calculations: $$ 0 \leq G(t) \leq 35 $$
Thus, the range of $G(t)$ is all real values from 0 to 35.
The range of $G(t)$ is all real numbers from $0$ to $35$.
More Information
The function reflects the linear relationship between time and the amount of data downloaded based on a constant download rate. Since the total game size is 35 gigabytes, the function cannot exceed this value.
Tips
- Confusing the maximum download time with the maximum data size. It's important to determine the range of the function based on the actual output values, not just the input time.
- Assuming the maximum upload time allows for downloading more than the total gigabytes available.
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