Difference between revisions of "2023 AMC 10A Problems/Problem 8"
m (→Solution 1) |
(I added solution 3, an easy to find, intuitive solution that is based on finding how much each Breadus unit is degrees Fahrenheit. This might be the wrong solution, but I did get the correct answer.) |
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~walmartbrian | ~walmartbrian | ||
− | ==Solution 2 ( | + | ==Solution 2 (Faster)== |
Let <math>^\circ B</math> denote degrees Breadus. We notice that <math>200^\circ F</math> is <math>90^\circ F</math> degrees to <math>0^\circ B</math>, and <math>150^\circ F</math> to <math>100^\circ B</math>. This ratio is <math>90:150=3:5</math>; therefore, <math>200^\circ F</math> will be <math>\dfrac3{3+5}=\dfrac38</math> of the way from <math>0</math> to <math>100</math>, which is <math>\boxed{\textbf{(D) }37.5.}</math> | Let <math>^\circ B</math> denote degrees Breadus. We notice that <math>200^\circ F</math> is <math>90^\circ F</math> degrees to <math>0^\circ B</math>, and <math>150^\circ F</math> to <math>100^\circ B</math>. This ratio is <math>90:150=3:5</math>; therefore, <math>200^\circ F</math> will be <math>\dfrac3{3+5}=\dfrac38</math> of the way from <math>0</math> to <math>100</math>, which is <math>\boxed{\textbf{(D) }37.5.}</math> | ||
~Technodoggo | ~Technodoggo | ||
+ | |||
+ | ==Solution 3 (Intuitive)== | ||
+ | |||
+ | From <math>110</math> to <math>350</math> degrees Fahrenheit, the Breadus scale goes from <math>1</math> to <math>100</math>. <math>110</math> to <math>350</math> degrees is a a span of <math>240</math>, and we can use this to determine how many Fahrenheit each Breadus unit is worth. <math>240</math> divided by <math>100</math> is <math>2.4</math>, so each Breadus unit is <math>2.4</math> Fahrenheit, starting at <math>110</math> Fahrenheit. For example, <math>1</math> degree on the Breadus scale is <math>110 + 2.4</math>, or <math>112.4</math> Fahrenheit. Using this information, we can figure out how many Breadus degrees <math>200</math> Fahrenheit is. <math>200-110</math> is <math>90</math>, so we divide <math>90</math> by <math>2.4</math> to find the answer, which is <math>\boxed{\textbf{(D) }37.5}</math> | ||
+ | |||
+ | ~MercilessAnimations | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2023|ab=A|num-b=7|num-a=9}} | {{AMC10 box|year=2023|ab=A|num-b=7|num-a=9}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 00:53, 10 November 2023
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
To solve this question, you can use where the
is the Fahrenheit and the
is the Breadus. We have
and
. We want to find
. 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 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
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.