[content] evaluation/easy_problems.jsonl (jsonl)
{"input": [{"role": "user", "content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.

When orange trees are planted to the acre, each tree yields 100 pounds of fruit. For each additional tree over 100, the yield per tree is reduced by 1 pound. How many trees should be planted so that the yield per acre is 4816 pounds?"}], "output": "<think>
Okay, let's try to figure out this problem. So, we're dealing with orange trees planted per acre and their yield. The problem says that when they're planted 100 per acre, each tree gives 100 pounds of fruit. But if you plant more than 100 trees, each additional tree over 100 causes the yield per tree to drop by 1 pound. We need to find how many trees should be planted to get a total yield of 4816 pounds per acre.

Hmm, let's break it down. Let me start by setting up variables. Let's say the number of trees per acre is x. If x is 100, then each tree gives 100 pounds, so total yield is 100*100 = 10,000 pounds. But since we want a yield of 4816, which is much less, maybe we need to plant more trees? Wait, but more trees mean each tree gives less. So there's a trade-off here.

The problem says for each additional tree over 100, the yield per tree decreases by 1 pound. So if we have x trees where x > 100, the number of extra trees is (x - 100). Therefore, the yield per tree would be 100 - (x - 100) pounds. Wait, let me check that. If you have x trees, and the extra trees over 100 are (x - 100), then each tree's yield is 100 minus that number. So, yield per tree = 100 - (x - 100) = 200 - x. Hmm, wait, that might not be right. Let me think again.

Wait, no. Let's say for each tree beyond 100, the yield decreases by 1. So if you plant 101 trees, each tree gives 100 - 1 = 99 pounds. If you plant 102 trees, each gives 98, and so on. So in general, if you plant x trees where x > 100, the yield per tree is 100 - (x - 100). Wait, but 100 - (x - 100) simplifies to 200 - x. So, yield per tree is (200 - x) pounds when x > 100. Then, the total yield would be x * (200 - x). So total yield is x*(200 - x). But we need this total yield to be 4816. So, the equation would be x*(200 - x) = 4816.

Wait, but let me make sure. Let me test with x=100. If x=100, then total yield would be 100*100 = 10,000, which matches the problem statement. If x=150, then yield per tree would be 100 - 50 = 50, so total yield 150*50=7500. That seems right.

So, according to this, the total yield when planting x trees (for x >= 100) is x*(200 - x). Wait, but hold on: when x=100, that gives 100*(200-100)=100*100=10,000, which is correct. For x=101, it's 101*(200 - 101)=101*99=9999, which aligns with each tree producing 1 less pound. So yes, that formula seems correct.

So, we set up the equation:

x*(200 - x) = 4816

But let me check, is 4816 possible? Let's see. The maximum possible yield is at x=100, which is 10,000. So 4816 is less than that. Wait, but when we go beyond 100 trees, each tree's yield decreases, but the number of trees increases. So maybe there's a point where the total yield is 4816. Wait, but maybe there are two such points? Because as x increases beyond 100, the total yield first decreases from 10,000, and maybe reaches a minimum and then... Wait, actually, the equation x*(200 - x) is a quadratic that opens downward, with vertex at x=100, which is the maximum point. Wait, no, x*(200 - x) is equivalent to -x^2 + 200x. The vertex is at x = -b/(2a) = -200/(2*-1) = 100. So yes, that's the maximum point. So as x increases beyond 100, the total yield decreases. Therefore, since 4816 is less than 10,000, there should be two solutions for x: one less than 100 and one greater than 100. But in the context of the problem, since the problem mentions \"additional trees over 100\", we're looking for x >= 100. Wait, but x can't be less than 100 here because the problem states that the reduction starts when you plant more than 100 trees. So if x is less than 100, does the yield stay at 100 pounds per tree? The problem says, \"When orange trees are planted to the acre\" (assuming that's 100 per acre?), \"each tree yields 100 pounds of fruit. For each additional tree over 100, the yield per tree is reduced by 1 pound.\" So if we plant fewer than 100 trees, does the yield per tree stay at 100? The problem doesn't specify, but since the question is about planting \"additional trees over 100\", implying that the base case is 100 trees. So maybe the model is that if you plant up to 100 trees, each gives 100 pounds, and beyond that, each additional tree reduces the yield by 1 per tree. So the total yield function would be:

If x <= 100, total yield = 100x

If x > 100, total yield = x*(200 - x)

But the problem wants the total yield to be 4816 pounds. Let's see if x <= 100 is possible. If x=48, 48*100=4800, which is close. x=49 would be 4900. But 4816 is between 4800 and 4900. But since 4816 isn't a multiple of 100, it can't be achieved with x <= 100. Therefore, the solution must be x > 100.

So the equation we need to solve is x*(200 - x) = 4816.

Let me write that as:

200x - x^2 = 4816

Rearranged to:

x^2 - 200x + 4816 = 0

Wait, moving everything to the left side:

-x^2 + 200x - 4816 = 0 --> multiply both sides by -1:

x^2 - 200x + 4816 = 0

Now, let's solve this quadratic equation for x.

We can use the quadratic formula:

x = [200 ± sqrt(200^2 - 4*1*4816)] / 2

Compute discriminant D:

D = 40000 - 4*4816 = 40000 - 19264 = 20736

sqrt(20736) = 144 (since 144*144=20736)

So,

x = [200 ± 144]/2

Calculating the two roots:

x = (200 + 144)/2 = 344/2 = 172

x = (200 - 144)/2 = 56/2 = 28

But earlier, we determined that x must be greater than 100, since the yield when x<=100 would be 100x, and 100*28=2800 which is less than 4816, and 100*48=4800, 100*49=4900. So 28 is too low and 172 is over 100. So x=172 is the solution.

Wait, but let me verify with x=172 trees. The yield per tree would be 100 - (172 - 100) = 100 - 72 = 28 pounds per tree. Then total yield is 172 * 28. Let's compute that.

172 * 28: 170*28=4760, 2*28=56, so 4760+56=4816. That's exactly the yield we need. So that works. The other solution x=28 would give 28 trees each at 100 pounds, which is 2800 pounds, which is not the desired 4816. Therefore, the answer is 172 trees.

So the answer should be 172. Let me just check once again. If 172 trees are planted, the yield per tree is 100 - (172 - 100) = 28, total 172*28=4816. Perfect. So yes, that's correct.
</think>

The number of trees to plant is found by solving the quadratic equation derived from the problem's conditions. Let \\( x \\) be the number of trees. For \\( x > 100 \\), the yield per tree is \\( 100 - (x - 100) = 200 - x \\). The total yield equation is:

\\[
x(200 - x) = 4816
\\]

Rewriting this as a quadratic equation:

\\[
x^2 - 200x + 4816 = 0
\\]

Using the quadratic formula \\( x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a} \\):

\\[
x = \\frac{200 \\pm \\sqrt{200^2 - 4 \\cdot 1 \\cdot 4816}}{2} = \\frac{200 \\pm 144}{2}
\\]

This yields two solutions: \\( x = 172 \\) and \\( x = 28 \\). Since \\( x \\) must be greater than 100 to affect the yield reduction, the valid solution is:

\\[
\\boxed{172}", "category": "math", "license": "cc-by-4.0", "reasoning": "on", "generator": "Qwen-2.5-32B-Instruct, DeepSeek-R1", "used_in_training": "Ultra, Super, Nano", "version": "v1.1", "system_prompt": "detailed thinking on"}