2024 AMC 10A Problems/Problem 2: Difference between revisions
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== | ==Video solution by TheNeuralMathAcademy== | ||
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Revision as of 12:03, 18 August 2025
- The following problem is from both the 2024 AMC 10A #2 and 2024 AMC 12A #2, so both problems redirect to this page.
Problem
A model used to estimate the time it will take a content creator to produce a custom video for a subscriber is of the form
![\[ T = aL + bG, \]](http://latex.artofproblemsolving.com/b/7/3/b732321f6682cbdbe08e24a43dbb888ddc9a692a.png)
where \( a \) and \( b \) are constants, \( T \) is the production time in minutes, \( L \) is the length of the video in minutes, and \( G \) is the number of special requests.
The model estimates that it will take \( 69 \) minutes to produce a video that is \( 1.5 \) minutes long with \( 800 \) special requests, as well as a video that is \( 1.2 \) minutes long with \( 1100 \) special requests.
How many minutes does the model estimate it will take to produce a \( 4.2 \)-minute video with \( 4000 \) special requests?
![\[ \textbf{(A) }240\qquad\textbf{(B) }246\qquad\textbf{(C) }252\qquad\textbf{(D) }258\qquad\textbf{(E) }264 \]](http://latex.artofproblemsolving.com/3/8/e/38e9ef91344b10186909d8b7a765c7a3ae06b075.png)
Solution 1
Plug in the values into the equation to give you the following two equations:
\begin{align*}
69&=1.5a+800b, \\
69&=1.2a+1100b.
\end{align*}
Solving for the values 
and 
gives you that 
and 
. These values can be plugged back in showing that these values are correct.
Now, use the given 
-mile length and 
-foot change in elevation, giving you a final answer of 
Solution by juwushu.
Solution 2
Alternatively, observe that using 
and 
makes the numbers much more closer to each other in terms of magnitude.
Plugging in the new variables: \begin{align*} 69&=15x+8y, \\ 69&=12x+11y. \end{align*}
The solution becomes more obvious in this way, with 
, and since 
, we determine that 
.
The question asks us for 
. Since 
, we have 
.
~
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https://youtu.be/l3VrUsZkv8I ~MC
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~IceMatrix
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Video Solution by yjtest
https://www.youtube.com/watch?v=kyDHwV_KEy4
