DeepSeek-R1 and Ollama3.2 - Judge Math

To find the area of the triangle with vertices at points (0, 0), (-1, 1), and (3, 3), we can use the formula for the area of a triangle given its vertices.

First, let's plot the points on a coordinate plane:

(0, 0)
(-1, 1)
(3, 3)

Next, we find the lengths of the base and height of the triangle. The length of the base can be found by calculating the distance between (-1, 1) and (3, 3), which is:

√((-1 - 3)^2 + (1 - 3)^2) = √((-4)^2 + (-2)^2) = √(16 + 4) = √20

The length of the height can be found by calculating the distance from (0, 0) to either point. Using the coordinates of (0, 0), we find:

√((3 - 0)^2 + (3 - 0)^2) = √(9 + 9) = √18

Since both base and height have a length of √20 and √18 respectively, we can use Heron's formula to calculate the area.

However, considering the points given, it seems there might be an easier approach that involves using vectors or a determinant method for finding the area.

To find the area using vectors, let's define the sides of the triangle as vectors:

Side 1: Vector from (-1, 1) to (3, 3) = <3 - (-1), 3 - 1> = <4, 2>
Side 2: Vector from (-1, 1) to (0, 0) = <-1 - 0, 1 - 0> = <-1, 1>

The area of the triangle can be found by calculating half the magnitude of the cross product of these two vectors.

Area = 1/2 |<4, 2> x <-1, 1>| = 1/2|4(-1) + 2(1)| = 1/2|-4 + 2| = 1/2|-2| = 1

Therefore, the area of the triangle with vertices at points (0, 0), (-1, 1), and (3, 3) is 1 square unit.

To find the area of the triangle, we can use the formula:

Area = (base × height) / 2

First, let's find the base of the triangle. We can do this by finding the distance between two points on the same line.

The x-coordinates of the vertices are 0 and -1, so the difference is 1 unit. The y-coordinates are 0 and 3, so the difference is also 3 units (but we need to subtract one from the other because the point (3, 3) lies above the line formed by points (0, 0) and (-1, -1)).

So base of triangle = 2 units

Now, let's find the height. The y-coordinates of the vertices are 1 and 3, so the difference is 2 units.

Height = 2 units

Now we can plug these values into the formula:

Area = (base × height) / 2
= (2 × 2) / 2
= 4/2
= 2

So the area of the triangle

To find the area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\), we can use the formula:

\[
\text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|
\]

For the given vertices \((0, 0)\), \((-1, 1)\), and \((3, 3)\), we assign:

- \(x_1 = 0\), \(y_1 = 0\)
- \(x_2 = -1\), \(y_2 = 1\)
- \(x_3 = 3\), \(y_3 = 3\)

Substitute these values into the formula:

\[
\text{Area} = \frac{1}{2} \left| 0(1-3) + (-1)(3-0) + 3(0-1) \right|
\]

Simplify each term:

- The first term: \(0 \times (1 - 3) = 0\)
- The second term: \((-1) \times (3 - 0) = -3\)
- The third term: \(3 \times (0 - 1) = -3\)

Combine these results:

\[
\text{Area} = \frac{1}{2} \left| 0 - 3 - 3 \right| = \frac{1}{2} \left| -6 \right|
\]

Calculate the absolute value and the area:

\[
\text{Area} = \frac{1}{2} \times 6 = 3
\]

Thus, the area of the triangle is \(\boxed{3}\).

## Step 1: Recall the formula to calculate the area of a triangle given its vertices
To find the area of the triangle formed by three points in the Cartesian plane, we can use the formula based on the determinant method. This involves calculating:

\[ \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \]

where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the triangle's vertices.

## Step 2: Assign the given points to their respective positions in the formula
Given that our points are (0, 0), (-1, 1), and (3, 3), we can assign these as follows:

- (x1, y1) = (0, 0)
- (x2, y2) = (-1, 1)
- (x3, y3) = (3, 3)

## Step 3: Plug the values into the formula
Substitute x1=0, y1=0; x2=-1, y2=1; and x3=3, y3=3 into the area formula:

\[ \text{Area} = \frac{1}{2} |0(1 - 3) + (-1)(3 - 0) + 3(0 - 1)| \]

## Step 4: Simplify inside the absolute value
Perform the arithmetic operations within the absolute value:

- \(0(1 - 3) = 0\)
- \((-1)(3 - 0) = -3\)
- \(3(0 - 1) = -3\)

So, inside the absolute value we have: \(0 - 3 - 3 = -6\).

## Step 5: Apply the absolute value and multiply by 1/2
Calculate the area using the simplified expression:

\[ \text{Area} = \frac{1}{2} |{-6}| \]

\[ \text{Area} = \frac{1}{2} \times 6 \]

## Step 6: Final calculation for the area
Compute the multiplication to find the final answer:

\[ \text{Area} = 3 \]

The final answer is: $\boxed{3}$

The startup invests $8000 in software development in the first year. In the second year, it invests half of that amount, which is $8000 / 2 = $4000.

So, for the first year, the total investment was $8000 + $4000 = $12,000.
For the second year, the total investment was $8,000 (half of the initial $16,000) + $4,000 = $12,000.

To find the total amount invested over the two years, add these amounts together: $12,000 + $12,000 = $24,000.

To find the total amount invested by the startup in software development over the two years, we need to calculate the investment for each year.

In the first year, the startup invests $8000.

In the second year, the startup invests half of that amount, which is:

$8000 / 2 = $4000

Now, let's add the investments from both years to find the total:

Total Investment = $8000 (Year 1) + $4000 (Year 2)
= $12,000

The total amount invested by the startup in software development over the two years is $12,000.

## Step 1: Determine the amount invested in the first year
The tech startup invests $8000 in software development in the first year.

## Step 2: Calculate the amount invested in the second year
In the second year, the startup invests half of the first-year's investment. Half of $8000 is $4000.

## Step 3: Sum the amounts invested over both years to find the total investment
To find the total amount invested, we need to add the investments from both years together. So, the total investment is $8000 (year one) + $4000 (year two).

## Step 4: Perform the addition to get the final answer
Adding the amounts calculated in step 3 gives us a total of $12000.

The final answer is: $\boxed{12000}$

To calculate the total amount invested by the startup in software development over the two years, we need to add the investments from each year.

1. In the first year, the startup invests $8,000.
2. In the second year, the investment is half of the first year's amount:
\[
\frac{8000}{2} = 4000
\]

Now, we add the investments from both years to find the total:

\[
8000 + 4000 = 12000
\]

Therefore, the total amount invested by the startup in software development over the two years is $12,000.

<think>
First, identify the investment made in each year. The startup invests \$8,000 in the first year.

In the second year, the startup decides to invest half of the initial amount. Half of \$8,000 is calculated as:

\$8,000 ÷ 2 = \$4,000

Next, sum up the investments from both years to find the total investment over two years:

\$8,000 (first year) + \$4,000 (second year) = \$12,000

Therefore, the startup invested a total of \$12,000 in software development over the two-year period.
</think>

To determine the total amount invested in software development over the two years, let's break down the problem step by step.

1. **First Year Investment:**
- **Amount Invested:** \$8,000
- **Total Investment After First Year:**

\[
\$8,\!000
\]

2. **Second Year Investment:**
- The startup decides to invest half of the first-year amount.
- Calculate half of \$8,000:

\[
\frac{\$8,\!000}{2} = \$4,\!000
\]
- **Total Investment After Second Year:**

\[
\$8,\!000 + \$4,\!000 = \$12,\!000
\]

3. **Final Total Investment:**
- Adding the investments from both years gives the total amount invested in software development over two years.

\[
\boxed{\$12,\!000}
\]

<think>
First, I identify the investments made by the tech startup in each year.

In the first year, the startup invested $8,000 in software development.

In the second year, they invested half of that amount. Calculating half of $8,000 gives $4,000.

To find the total investment over the two years, I add both amounts together: $8,000 + $4,000 = $12,000.
</think>

**Solution:**

Let's break down the investments year by year.

1. **First Year Investment:**

The startup invested \$8,\!000 in software development during the first year.

2. **Second Year Investment:**

In the second year, the startup invested half of the first year's amount.

\[
\text{Second Year Investment} = \frac{\$8,\!000}{2} = \$4,\!000
\]

3. **Total Investment Over Two Years:**

To find the total investment, add both years' investments together.

\[
\text{Total Investment} = \$8,\!000 + \$4,\!000 = \$12,\!000
\]

**Final Answer:**

\[
\boxed{\$12,\!000}
\]

<think>
To determine the total investment made by the tech startup in software development over the two years, I will break down each year's investment separately.

First Year Investment:
The company invests \$8,000 in software development during the first year.

Second Year Investment:
In the second year, the company invests half of the amount invested in the first year. So, half of \$8,000 is \$4,000.

Total Investment:
To find the total investment over both years, I will add the investments from each year together: \$8,000 (first year) + \$4,000 (second year).

This calculation shows that the tech startup invested a total of \$12,000 in software development over the two years.
</think>

**Solution:**

Let's calculate the total investment made by the tech startup in software development over the two years.

1. **First Year Investment:**
- The startup invests **\$8,000** in software development during the first year.

2. **Second Year Investment:**
- In the second year, the startup invests half of the amount invested in the first year.
- Half of \$8,000 is:
\[
\frac{1}{2} \times 8,\!000 = 4,\!000
\]

3. **Total Investment Over Two Years:**
- To find the total investment over both years, add the investments from each year.
- Total Investment:
\[
\$8,\!000 + \$4,\!000 = \$12,\!000
\]

**Final Answer:**
The tech startup invested a total of \boxed{\$12,\!000} in software development over the two years.

<think>
First, I need to determine the investment made by the startup in each of the two years.

In the first year, the startup invests $8000 in software development.

For the second year, the startup invests half of the amount it invested in the first year. Half of $8000 is $4000.

To find the total investment over the two years, I add the amounts from both years together: $8000 + $4000 equals $12000.
</think>

Let's calculate the total amount invested by the startup in software development over the two years step by step.

**First Year Investment:**
- **Amount Invested:** \$8,000

**Second Year Investment:**
- **Investment for Second Year:** Half of the first year's investment
- **Calculation:**
\[
\frac{1}{2} \times \$8,000 = \$4,000
\]

**Total Investment Over Two Years:**
- **First Year:** \$8,000
- **Second Year:** \$4,000
- **Total Amount Invested:**
\[
\$8,000 + \$4,000 = \boxed{\$12,\!000}
\]

**Final Answer:**
\[
\boxed{\$12,\!000}
\]

<think>
First, identify the investments for each year.

In the first year, the startup invests $8,000 in software development.

In the second year, they invest half of the first year's investment. So, I'll calculate 50% of $8,000 to find the second-year investment.

Finally, add both years' investments together to determine the total amount invested over the two years.
</think>

**Solution:**

Let's break down the problem step by step to find the total amount invested in software development over the two years.

1. **First Year Investment:**
- The startup invests **\$8,000** in the first year.

2. **Second Year Investment:**
- In the second year, they invest **half of that amount**.
- To find half of \$8,000:
\[
\frac{1}{2} \times 8,\!000 = 4,\!000
\]
- So, the investment in the second year is **\$4,000**.

3. **Total Investment Over Two Years:**
- Add both years' investments:
\[
8,\!000 + 4,\!000 = 12,\!000
\]

**Final Answer:**
\[
\boxed{12,\!000}
\]

## Step 1: Determine the total percentage of students who prefer either blue or green.
First, let's add up the percentages of students who like blue and those who like green. However, since we're double-counting the ones who like both colors, we need to subtract that percentage once.

## Step 2: Calculate the combined percentage of students who like blue or green.
Percentage for blue = 58%
Percentage for green = 45%
Percentage for both blue and green = 22%

Combined percentage (blue or green) = Percentage for blue + Percentage for green - Percentage for both blue and green
= 58% + 45% - 22%
= 81%

## Step 3: Calculate the percentage of students who like neither blue nor green.
To find this, we subtract the combined percentage of students who prefer either color from 100%.

Percentage of students who like neither blue nor green = 100% - Combined percentage (blue or green)
= 100% - 81%
= 19%

## Step 4: Convert the percentage into a decimal for probability calculation.
Since probability is calculated using decimals, we'll convert 19% to a decimal by dividing it by 100.

Probability that they would like neither blue nor green = 19 / 100
= 0.19

The final answer is: $\boxed{0.19}$

To solve this problem, we can use the principle of inclusion-exclusion to find the probability that a student likes either blue or green or both.

Let:
- \( P(B) \) be the probability that a student likes blue.
- \( P(G) \) be the probability that a student likes green.
- \( P(B \cap G) \) be the probability that a student likes both blue and green.

From the problem, we have:
- \( P(B) = 0.58 \)
- \( P(G) = 0.45 \)
- \( P(B \cap G) = 0.22 \)

We want to find the probability that a student likes either blue or green or both, which is given by:

\[
P(B \cup G) = P(B) + P(G) - P(B \cap G)
\]

Substitute the given values:

\[
P(B \cup G) = 0.58 + 0.45 - 0.22 = 0.81
\]

This means that 81% of students like either blue or green or both.

To find the probability that a student likes neither color, we calculate the complement of \( P(B \cup G) \):

\[
P(\text{neither}) = 1 - P(B \cup G) = 1 - 0.81 = 0.19
\]

Thus, the probability that a randomly picked student would like neither blue nor green is \( \boxed{0.19} \).

<think>
To find the probability that a randomly selected student likes either blue or green or both, I'll use the principle of inclusion-exclusion.

First, I know that 58% of students liked blue and 45% preferred green. However, some students liked both colors, which is 22%.

Using the formula for the union of two sets, P(Blue ∪ Green) = P(Blue) + P(Green) - P(Both). Plugging in the numbers: 0.58 + 0.45 - 0.22 equals 0.81.

This means 81% of students liked either blue or green. To find those who liked neither, I subtract this probability from 1: 1 - 0.81 = 0.19.

So, the probability that a randomly picked student likes neither blue nor green is 19%.
</think>

To determine the probability that a randomly selected student **likes neither** blue nor green, we can use the principle of **inclusion-exclusion**. Here's a step-by-step solution:

1. **Define the Given Probabilities:**
- Let \( P(Blue) = 58\% \)
- Let \( P(Green) = 45\% \)
- Let \( P(\text{Both Blue and Green}) = 22\% \)

2. **Calculate the Probability of Liked Either Blue or Green (or Both):**

Using the inclusion-exclusion principle:
\[
P(Blue \cup Green) = P(Blue) + P(Green) - P(\text{Both})
\]

Plugging in the numbers:
\[
P(Blue \cup Green) = 58\% + 45\% - 22\% = 81\%
\]

3. **Determine the Probability of Liked Neither Blue nor Green:**

The probability that a student likes neither color is the complement of liking either blue or green:
\[
P(\text{Neither}) = 100\% - P(Blue \cup Green)
\]

Substituting the value we found:
\[
P(\text{Neither}) = 100\% - 81\% = 19\%
\]

**Final Answer:**
\[
\boxed{19\%}
\]

To find the probability that a randomly selected student likes neither blue nor green, we first need to find the probability of liking either blue or green.

We know that 58% liked blue (B) and 45% liked green (G). Since these two events are not mutually exclusive, we can use the formula for the union of two events:

P(B ∪ G) = P(B) + P(G) - P(B ∩ G)

where ∩ represents "and".

Given that 22% liked both colors, this means P(B ∩ G) is equal to 22%. We also know that P(B) = 58%, P(G) = 45%.

Now we can plug these values into the formula:

P(B ∪ G) = P(B) + P(G) - P(B ∩ G)
= 58% + 45% - 22%
= 115%

So, the probability of a randomly selected student liking either blue or green is 115%. Now to find the probability that they like neither color we use the complement rule:

P(neither B nor G) = 1 - P(B ∪ G)
= 1 - 115%
= 0.85

To find the probability that a randomly chosen student likes neither blue nor green, we need to first find the total percentage of students who like either color.

We are given:
- 58% of students liked blue (B)
- 45% preferred green (G)
- 22% liked both colors (B ∩ G)

Using the principle of inclusion-exclusion, we can calculate the total percentage of students who like at least one color as follows:

P(B or G) = P(B) + P(G) - P(B ∩ G)
= 0.58 + 0.45 - 0.22
= 0.81

This means that 81% of students like either blue, green, or both colors.

To find the probability that a student likes neither color, we subtract this percentage from 100%:

P(neither B nor G) = 1 - P(B or G)
= 1 - 0.81
= 0.19

So, the probability that a randomly chosen student at the school would like neither blue nor green is 19%.