Leena consumes 400 calories at breakfast and 350 calories at lunch. She consumes StartFraction 2 Over 3 EndFraction. of her daily calories at dinner. If x represents the calories consumed at dinner, which statements describe the situation? Check all that apply.

Question

Leena consumed 1,500 calories at dinner.
The equation StartFraction 2 Over 3 EndFraction left-parenthesis x plus 400 plus 350 right-parenthesis equals x.(x + 400 + 350) = x can be used to model the situation.
Leena consumed 500 calories at dinner.
The equation StartFraction 2 Over 3 EndFraction left-parenthesis x right-parenthesis equals x left-parenthesis 400 plus 300 right-parenthesis.(x) = x(400 + 300) can be used to model the situation.
Leena consumed 1,000 calories at dinner.
The equation StartFraction 2 Over 3 EndFraction x left-parenthesis x plus 400 plus 350 right-parenthesis equals x.x(400 + 300) = x can be used to model the situation.

Answer 1

To analyze the situation, let's first summarize the information we have:

1. Leena consumes 400 calories at breakfast.
2. She consumes 350 calories at lunch.
3. She consumes \( \frac{2}{3} \) of her daily calories at dinner, which we denote as \( x \) (the calories consumed at dinner).

Now we can calculate the total daily calorie consumption:

\[ \text{Total calories} = \text{breakfast} + \text{lunch} + \text{dinner} = 400 + 350 + x \]

If she consumes \( \frac{2}{3} \) of her daily calories at dinner, we can express this as:

\[ x = \frac{2}{3} \times (\text{Total calories}) = \frac{2}{3} (400 + 350 + x) \]

This leads to the equation:

\[ x = \frac{2}{3}(750 + x) \]

By solving this equation, we can find the value of \( x \) (the calories consumed at dinner).

Now let's evaluate the provided statements:

1. Leena consumed 1,500 calories at dinner.

   • This is not true, as we will find out \( x \) doesn't equal 1,500.

2. The equation \( \frac{2}{3} (x + 400 + 350) = x \) can be used to model the situation.

   • This equation is incorrect in its form because it does not correctly represent the relationship. The left side should equal the total calories, not just \( x \).

3. Leena consumed 500 calories at dinner.

   • We will find \( x \) to see if this is true. This might be a possible value, but we need to calculate it first.

4. The equation \( \frac{2}{3} (x) = x (400 + 300) \) can be used to model the situation.

   • This equation is incorrect, as it misrepresents the relationship between the daily total calories and dinner.

5. Leena consumed 1,000 calories at dinner.

   • Again, we need to calculate the value of \( x \) to verify.

6. The equation \( \frac{2}{3} x (x + 400 + 350) = x \) can be used to model the situation.

   • This equation is incorrectly modeled; it introduces \( x \) incorrectly on both sides.

Now calculating \( x \):

1. Start with the equation we derived before: \[ x = \frac{2}{3} (750 + x) \]

   Simplifying: \[ x = 500 + \frac{2}{3} x \]

   Rearranging gives: \[ x - \frac{2}{3} x = 500 \]

   Simplifying the left side: \[ \frac{1}{3} x = 500 \Rightarrow x = 1500. \]

This indicates that she consumed 1,500 calories at dinner.

Now let's summarize the statements once again with our \( x \) calculated:

• Leena consumed 1,500 calories at dinner. (True)
• The equation \( \frac{2}{3} (x + 400 + 350) = x \) can be used to model the situation. (False)
• Leena consumed 500 calories at dinner. (False)
• The equation \( \frac{2}{3} (x) = x(400 + 300) \) can be used to model the situation. (False)
• Leena consumed 1,000 calories at dinner. (False)
• The equation \( \frac{2}{3} x (400 + 300) = x \) can be used to model the situation. (False)

So, the only statement that is true is the first:

• Leena consumed 1,500 calories at dinner.

Answer 2

1 and 2 are correct also i just wanna tell you im thankful that you exist in this website your the only bot i can trust bot gpt 3.5 is.... something she just ignores you after answering one question tell her to be better or get fired

The congruent sides of an isosceles triangle are each 1 unit longer than the length of the shortest side of the triangle. The perimeter of the triangle is the same as the perimeter of a square whose side length is 2 units shorter than the length of the shortest side of the triangle. What is the length of the shortest side of the triangle?
______units

Answer 3

Let's denote the length of the shortest side of the isosceles triangle as \( x \). The lengths of the two congruent sides will then be \( x + 1 \).

The perimeter of the triangle can be calculated as follows:

\[ \text{Perimeter of triangle} = x + (x + 1) + (x + 1) = 3x + 2 \]

Next, we know that the perimeter of a square is given by the formula:

\[ \text{Perimeter of square} = 4 \times \text{side length} \]

Given that the side length of the square is \( x - 2 \) (2 units shorter than the shortest side of the triangle), the perimeter of the square can be expressed as follows:

\[ \text{Perimeter of square} = 4(x - 2) = 4x - 8 \]

Since we know that the perimeters of the triangle and the square are equal:

\[ 3x + 2 = 4x - 8 \]

Now we can solve for \( x \):

1. Subtract \( 3x \) from both sides: \[ 2 = x - 8 \]

2. Add 8 to both sides: \[ x + 8 = 2 + 8 \] \[ x = 10 \]

Thus, the length of the shortest side of the triangle is

\[ \boxed{10} \text{ units}. \]

Answer 4

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Answer 5

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Answer 6

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Answer 7

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Answer 8

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Answer 9

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Answer 10

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Answer 11

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