AMC 8, 2024 Problem Marathon

[Manya Lohumi]

Problem - 10

Answer - number of years between 1980 to 2030 = 50

As the ppm rate increases by 1.1515 ppm per year

So the ppm rate in 2030 will be 50 × 1.1515 = 57.575

338 + 57.575 = 395.575

Rounded to the nearest given integer = 399

The ppm rate is supposed to be 399.

Option - ( A ) - 399

 

 

[Arisha Roy]

Problem 21

A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and turns yellow when in the sun. Initially, the ratio of green to yellow frogs was \(3: 1\). Then 3 green frogs moved to the sumny side and 5 yellow frogs moved to the shady side. Now the ratio is \(4: 1\). What is the difference between the number of green frogs and the number of yellow frogs now?

 

green frogs : yellow frogs \(=3: 1\)
Let green frogs \(=3 x\), yellow frogs \(=x\)

Now,
\[
\begin{aligned}
& \frac{3 x-3+5}{x+3-5}=\frac{4}{1} \\
& \frac{3 x+2}{x-2}=\frac{4}{1} \\
& 3 x+2=4 x-8 \\
& x=10
\end{aligned}
\]

Previously, green frogs \(=30\), yellow frogs \(=10\)
Now, green frogs \(=30-3+5=32\)
yellow frogs \(=10+3-5=8\)
Difference between the number of green frogs and the number of yellow frogs now \(=32-8=24\)

 

 

• This reply was modified 1 year, 5 months ago by Arisha Roy.

[Manya Lohumi]

Problem 4

Answer -

The sum of numbers from 1 to 9 is - 1+2+3+4+5+6+7+8+9= 45

We will find the answer by substracting the options from 45. The square will be the answer.

45 - 5 = 40  -    not a square

45 - 6 = 39  -    not a square

45 - 7 = 38  -    not a square

45 - 8 = 37  -    not a square

45 - 9 = 36  -    is a square

As 36 is a square number and it came by substracting 9 from 45

9 was the number which unji left out.

Option - ( E ) - 9

 

[Arisha Roy]

Problem 24

Jean has made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is 8 feet high while the other peak is 12 feet high. Each peak forms a \(90^{\circ}\) angle, and the straight sides form a \(45^{\circ}\) angle with the ground. The artwork has an area of 183 square feet. The sides of the mountain meet at an intersection point near the center of the artwork, \(h\) feet above the ground. What is the value of \(h\) ?

[Arisha Roy]

Extending the inner part of the mountain in the given figure,

This is the image of two right angle triangles that overlap in a third right angle triangle.
For the largest right angle triangle, other angles except the right angle is \(45^{\circ}\) and \(45^{\circ}\). The height from the right angle to the hypotenuse is 12 feet.
So, 2 sides of the triangle are similar and each side will be \(12\sqrt{2}\) feet.
So, the area will be \(144 \text{feet}^2\).
Similarly, each of the similar side of the second largest triangle will be \(8\sqrt{2}\) and area will be \(64 \text{feet}^2\).
The smallest triangle is a right angle triangle and each side is \(h\sqrt{2}\) feet.
So, the area will be \(h^2\).
If we add the area of the two triangles the area of the smallest triangle will be counted twice.
\(144+64-h^2=183\)
\(\Rightarrow h^2 = 25\)
\(\Rightarrow h=5\)
So, the value of h is 5 feet.

• This reply was modified 1 year, 5 months ago by Arisha Roy.

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