Responses
an=80,000(0.06)n−1
a subscript n baseline equals 80,000 left parenthesis 0.06 right parenthesis superscript n minus 1 baseline
an=80,000(0.06)n
a subscript n baseline equals 80,000 left parenthesis 0.06 right parenthesis superscript n baseline
an=80,000(0.94)n
a subscript n baseline equals 80,000 left parenthesis 0.94 right parenthesis superscript n baseline
an=80,000(0.94)n−1
9 answers
To model the depreciation of the bakery's equipment using a sequence, we need to account for the equipment's value decreasing by 6 percent per year. This means that each year, the equipment retains 94 percent of its value (100% - 6% = 94%).
Thus, the value of the equipment after \( n \) years can be modeled by the equation:
\[ a_n = 80,000(0.94)^n \]
Therefore, the correct equation to use is:
\( a_n = 80,000(0.94)^n \)
To model the decline of the herd of elephants, we need to account for a decrease of 4% each year. This means that each year, the population retains 96% of its previous population (100% - 4% = 96%).
Thus, the exponential decay function for the number of elephants after \( t \) years can be expressed as:
\[ y = 62(0.96)^t \]
So, the correct equation to model the problem is:
\( y = 62(0.96)^t \)
To determine the correct graph for the exponential decay modeled by the equation \( y = 4,060(0.95)^t \), we should first calculate the number of birds at the specified years (0, 5, 10, 15, and 20).
Here are the calculations:
At \( t = 0 \): \[ y = 4,060(0.95)^0 = 4,060 \times 1 = 4,060 \]
At \( t = 5 \): \[ y = 4,060(0.95)^5 \approx 4,060 \times 0.77378 \approx 3,141.12 \]
At \( t = 10 \): \[ y = 4,060(0.95)^{10} \approx 4,060 \times 0.61391 \approx 2,487.48 \]
At \( t = 15 \): \[ y = 4,060(0.95)^{15} \approx 4,060 \times 0.46329 \approx 1,882.83 \]
At \( t = 20 \): \[ y = 4,060(0.95)^{20} \approx 4,060 \times 0.34868 \approx 1,415.93 \]
Now, compiling the results in a table:
Years (t) | Number of Birds (y)
------------------------------
0 | 4,060
5 | ~3,141
10 | ~2,487
15 | ~1,883
20 | ~1,416
Now, we look at the responses provided to match this sequence:
The first response shows a decreasing pattern ranging from 0 to 600 on the y-axis, which is not suitable for our values.
The second response has a y-axis from 0 to 6000, which is also not suitable.
The third response portrays a decreasing pattern from around 40,000 to 52,000, which does not correspond to our calculated values.
The fourth response shows a decreasing curve with a y-axis from 0 to 6000, which, although not exact, could potentially encompass values around \( 1,416 \).
Given the calculations, the correct response would most likely be the fourth option, as it shows a decreasing trend and can cover the calculated values, contingent on the axis scaling.
Without seeing the exact graphs, it's difficult to give a definitive answer, but based on the information provided, the fourth graph appears to be the most appropriate choice.
In the context of exponential decay functions, the horizontal asymptote represents the value that the function approaches as \( t \) (or any independent variable) approaches infinity. Generally, for standard forms of exponential decay, the horizontal asymptote is typically at \( y = 0 \), meaning that the value of the function gets closer and closer to zero but never actually reaches it.
Given the options provided:
The most common horizontal asymptote for an exponential decay function is at \( y = 0 \).
Thus, if we are speaking about a standard exponential decay function without any other transformations affecting the baseline horizontally, the correct answer is:
The horizontal asymptote is at \( y = 0 \).
To determine which graph has the properties specified:
Y-intercept at (0, 22): This means that when \( x = 0 \), the value of \( y \) should be 22.
Horizontal asymptote at \( y = 5 \): This indicates that as \( x \) approaches infinity (or negative infinity), the graph approaches \( y = 5 \).
The graph is decreasing for all values of \( x \): A decreasing graph means that as \( x \) increases, \( y \) decreases.
Let's analyze the provided options based on these criteria:
Graph 1: The y-axis ranges from -2 to 26, and there is a solid curve graph. We need to see if it intersects at (0, 22) and decreases toward \( y = 5 \).
Graph 2: Similar to the first, but it has a range that goes up to 30. Again, check the y-intercept and the horizontal asymptote.
Graph 3: The y-axis ranges from -2 to 24, with a solid curve that intersects the y-axis. We would check if this curve intersects at (0, 22) and decreases toward \( y = 5\).
Graph 4: Here the y-axis ranges from -2 to 26. Again, we need to see if the curve meets the defined y-intercept and behavior.
To meet all the criteria:
If you can assess the images based on these characteristics, you would be looking for the graph that matches having a y-intercept at (0, 22), a horizontal asymptote at \( y = 5 \), and shows a continual decrease.
If I had to deduce from the descriptions:
Hence, without the capability to view the images directly, I recommend choosing the graph that shows a y-intercept at (0, 22) and can visually be confirmed to have a horizontal asymptote of \( y = 5 \) while consistently decreasing.