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Which of the following values for n makes the inequality 2n−−√<5

true? (1 point)
Responses

8
8

12.5
12.5

18
18

16

5 answers

To solve the inequality \( \sqrt{2n} < 5 \), we first square both sides to eliminate the square root. This gives us:

\[ 2n < 25 \]

Next, we divide both sides by 2:

\[ n < 12.5 \]

Now, we can evaluate the given options:

  • 8: \( 8 < 12.5 \) — True
  • 12.5: \( 12.5 < 12.5 \) — False
  • 18: \( 18 < 12.5 \) — False
  • 16: \( 16 < 12.5 \) — False

The only value for \( n \) that makes the inequality true is 8.

Get ready for an amusement park day of fun! You and your family want to spend the day at a theme park. You have pooled your money together and have $685. The tickets to the park are $125 each and there is an online processing fee of $5.75 per ticket. You also need to pay $20 for parking. If you have a family of five, do you have enough money to purchase these tickets? Find an inequality to determine how many tickets can be purchased. Then, find how many tickets you are able to purchase based on your inequality.(1 point)
Responses

125t+5.75t+20≤685
, and yes, you can purchase the tickets.
125 t plus 5 point 7 5 t plus 20 is less than or equal to 685, and yes, you can purchase the tickets.

125t+5.75t≤685
, and yes, you can purchase the tickets.
125 t plus 5 point 7 5 t is less than or equal to 685, and yes, you can purchase the tickets.

125t+5.75t+20≤685
, and no, you cannot purchase the tickets.
125 t plus 5 point 7 5 t plus 20 is less than or equal to 685, and no, you cannot purchase the tickets.

125t≤685
, and yes, you can purchase the tickets.

To determine whether you have enough money to purchase tickets for a family of five, we need to set up an inequality based on the total cost of the tickets, the processing fees, and the parking fee.

Let \( t \) represent the number of tickets. The total cost to purchase tickets can be broken down as follows:

  1. The cost of the tickets: \( 125t \)
  2. The total online processing fee: \( 5.75t \)
  3. The parking fee: \( 20 \)

Putting this all together, the inequality for the total cost can be expressed as:

\[ 125t + 5.75t + 20 \leq 685 \]

Now, we can combine the terms involving \( t \):

\[ (125 + 5.75)t + 20 \leq 685 \]

Calculating \( 125 + 5.75 \):

\[ 130.75t + 20 \leq 685 \]

Next, we isolate \( t \):

\[ 130.75t \leq 685 - 20 \]

\[ 130.75t \leq 665 \]

Now, divide both sides by \( 130.75 \):

\[ t \leq \frac{665}{130.75} \approx 5.08 \]

Since you can't purchase a fraction of a ticket, the maximum number of tickets you can purchase is 5.

Now let's evaluate the options given:

  1. 125t + 5.75t + 20 ≤ 685, and yes, you can purchase the tickets.
  2. 125t + 5.75t ≤ 685, and yes, you can purchase the tickets.
  3. 125t + 5.75t + 20 ≤ 685, and no, you cannot purchase the tickets.
  4. 125t ≤ 685, and yes, you can purchase the tickets.

The correct response is:

125t + 5.75t + 20 ≤ 685, and yes, you can purchase the tickets.

The Ramirez family is going to the space museum. Adult tickets are $15 and children's tickets are $12. They want to spend a maximum of $150. Write a two-variable inequality to model this situation and determine the contextual constraints. Show your work as necessary.(1 point)

To model the situation involving the Ramirez family's ticket purchases for the space museum, we will define two variables:

  • Let \( x \) be the number of adult tickets purchased.
  • Let \( y \) be the number of children's tickets purchased.

Step 1: Set up the inequality

The cost for adult tickets is $15 each, and the cost for children's tickets is $12 each. The total amount they want to spend must not exceed $150. Therefore, we can write the inequality as:

\[ 15x + 12y \leq 150 \]

Step 2: Identify the contextual constraints

The context of the problem imposes certain constraints:

  1. Non-negative ticket constraints: The family cannot purchase a negative number of tickets. Therefore:

    • \( x \geq 0 \) (number of adult tickets)
    • \( y \geq 0 \) (number of children's tickets)

  2. Maximum spending constraint: The total cost should not exceed $150, which is already included in our inequality.

Summary of the two-variable inequality and constraints:

  • Inequality: \[ 15x + 12y \leq 150 \]

  • Constraints: \[ x \geq 0 \quad \text{(adult tickets)} \] \[ y \geq 0 \quad \text{(children's tickets)} \]

Interpretation

This inequality and the constraints ensure that the Ramirez family can determine the combinations of adult and children's tickets to purchase without exceeding their budget of $150.

You can also find combinations of \( x \) and \( y \) that satisfy the inequality by substituting whole numbers for \( x \) and solving for \( y \), ensuring that both \( x \) and \( y \) remain non-negative.