2023 AMC 10A Problems/Problem 8
Contents
- 1 Problem
- 2 Solution 1 (Substitution)
- 3 Solution 2 (Faster)
- 4 Solution 3 (Intuitive)
- 5 Solution 4
- 6 Solution 5
- 7 Solution 6 (extremely simple)
- 8 Video Solution by Math-X (First understand the problem!!!)
- 9 Video Solution by CosineMethod [🔥Fast and Easy🔥]
- 10 Video Solution
- 11 Video Solution (easy to digest) by Power Solve
- 12 See Also
Problem
Barb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function of the Fahrenheit scale. Bread rises at degrees Fahrenheit, which is degrees on the Breadus scale. Bread is baked at degrees Fahrenheit, which is degrees on the Breadus scale. Bread is done when its internal temperature is degrees Fahrenheit. What is this in degrees on the Breadus scale?
Solution 1 (Substitution)
To solve this question, you can use where the is Fahrenheit and the is Breadus. We have and . We want to find the value of in that falls on this line. The slope for these two points is ; . Solving for using , . We get . Plugging in . Simplifying,
~walmartbrian
Solution 2 (Faster)
Let denote degrees Breadus. We notice that is degrees to , and to . This ratio is ; therefore, will be of the way from to , which is
~Technodoggo
Solution 3 (Intuitive)
From to degrees Fahrenheit, the Breadus scale goes from to . to degrees is a span of , and we can use this to determine how many Fahrenheit each Breadus unit is worth. divided by is , so each Breadus unit is Fahrenheit, starting at Fahrenheit. For example, degree on the Breadus scale is , or Fahrenheit. Using this information, we can figure out how many Breadus degrees Fahrenheit is. is , so we divide by to find the answer, which is
~MercilessAnimations
Solution 4
We note that the range of F temperatures that represents is . is along the way to getting to , the end of this range, or of the way. Therefore if we switch to the Br scale, we are of the way to from , or at .
~Dilip -missmango ~Minor edits by FutureSphinx
Solution 5
We have the points and . We want to find . The equation of the line is . We use this to find , or . ~MC413551
Solution 6 (extremely simple)
We can write the value on the Breadus scale as , where is the temperature in Fahrenheit. From the problem, and We can rewrite this problem in terms of linear algebra to solve it.
We can write the system of equations as Ax = B. We can solve for x using the expression x = . Calculating this value we get Therefore, . Plugging in , we get .
~Loquacious Autodidact
Video Solution by Math-X (First understand the problem!!!)
https://youtu.be/GP-DYudh5qU?si=prG8ONR_AgTR4HkL&t=1683 ~Math-X
Video Solution by CosineMethod [🔥Fast and Easy🔥]
https://www.youtube.com/watch?v=dfrF_P-FIEA
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution (easy to digest) by Power Solve
https://www.youtube.com/watch?v=Yi5p3_x9iU8
See Also
2023 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.