1. Maria has 8 cups of a fruit punch that consists of 25% orange juice and 75% apple juice. She wants to make a drink that is 40% orange juice. How many cups of orange juice should Maria add to th mixture?
A: ?
Determine whether the trinomial is a perfect-square. If so, factor. If not, explain.
2. 16x^2 - 24x - 9
A: ?
(I think this trinomial may be a perfect-square, but my textbook tells me otherwise.
1. To solve this problem, we need to determine the amount of orange juice in the original mixture and calculate how much more should be added to achieve a 40% concentration.
Step 1: Find the amount of orange juice in the original mixture.
- The original mixture consists of 8 cups of fruit punch, which is composed of 25% orange juice.
- So, the amount of orange juice in the original mixture is 8 cups * 0.25 = 2 cups.
Step 2: Calculate the desired amount of orange juice in the final mixture.
- We want the final mixture to have a concentration of 40% orange juice.
- Let's assume Maria adds x cups of orange juice to the mixture.
- So, the desired amount of orange juice in the final mixture is (2 cups + x cups).
Step 3: Set up the equation.
- We know that the desired amount of orange juice is (2 cups + x cups), and this should constitute 40% of the total volume of the final mixture.
- Thus, the equation can be written as: (2 cups + x cups) = 0.4 * (8 cups + x cups).
- Simplifying the equation:
2 cups + x cups = 3.2 cups + 0.4x cups.
Step 4: Solve for x.
- Subtracting 0.4x cups from both sides:
2 cups + x cups - 0.4x cups = 3.2 cups.
- Simplifying the equation further:
0.6x cups = 1.2 cups.
- Dividing both sides by 0.6:
x cups = 1.2 cups / 0.6.
- Calculating:
x cups = 2 cups.
Therefore, Maria should add 2 cups of orange juice to the mixture.
2. To determine whether the trinomial 16x^2 - 24x - 9 is a perfect square, we can follow these steps:
Step 1: Check the coefficient of the terms.
- The coefficient of the x^2 term is 16, which is a perfect square since √16 = 4.
- The coefficient of the x term, -24, is not a perfect square.
Step 2: Look at the constant term.
- The constant term in this case is -9, which is also not a perfect square.
Therefore, the trinomial 16x^2 - 24x - 9 is not a perfect square.
(4x-3)^2 = 16x^2 - 24x + 9
You have -9. A perfect square will never end in a negative number.