State whether the equation represents exponential​ growth, exponential​ decay, or neither.

This equation represents exponential growth.

At a local fitness​ center, members pay an ​$8 membership fee and ​$5 for each aerobics class. Nonmembers pay ​$6 for each aerobics class. For what number of aerobics classes will the cost for members and nonmembers be the​ same?

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Part 1
​Thus, for
  
enter your response here aerobics​ classes, the cost will be the same for members and nonmembers.
​(Type a whole​ number.)

For what number of aerobics classes will the cost for members and nonmembers be the same?

Let x be the number of aerobics classes.

For members: Cost = $8 + $5x
For nonmembers: Cost = $6x

So, we need to find the value of x for which $8 + $5x = $6x.

$8 + $5x = $6x
$8 = $6x - $5x
$8 = $x

Therefore, for 8 aerobics classes, the cost will be the same for members and nonmembers.

Tell whether the system has one​ solution, infinitely many​ solutions, or no solution.

10yequalsminus4xplus20
2xplus5yequals10
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Part 1
Choose the correct answer below.
A.
The system has no solution.
B.
The system has infinitely many solutions.
C.
The system has one solution.

The system of equations:

10y = -4x + 20
2x + 5y = 10

has one solution.

This can be determined since the system has two linear equations with two variables, which can be represented graphically as two lines that intersect at a unique point, representing the solution to the system.

Gabe​ Amodeo, a nuclear​ physicist, needs 50 liters of a 60​% acid solution. He currently has a 20 % solution and a 70 % solution. How many liters of each does he need to make the needed 50 liters of 60​% acid​ solution?

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Part 1
Gabe needs
  
enter your response here liters of the 20​% solution.
Part 2
He also needs
  
enter your response here liters of the 70​% solution.

Let's represent the amount of 20% solution Gabe needs with x liters and the amount of 70% solution with y liters.

The total amount of the 60% acid solution needed is 50 liters.

We can set up the following system of equations based on the information given:

1. x + y = 50 (Total liters of solution)
2. 0.20x + 0.70y = 0.60(50) (Total acid content required)

Solving the system of equations:
From equation 1, we can express x in terms of y:
x = 50 - y

Substitute x = 50 - y into equation 2 and solve for y:
0.20(50 - y) + 0.70y = 0.60(50)
10 - 0.20y + 0.70y = 30
0.50y = 20
y = 40

Now, substitute y = 40 back into x = 50 - y to find x:
x = 50 - 40
x = 10

Therefore, Gabe needs 10 liters of the 20% solution and 40 liters of the 70% solution to make the needed 50 liters of 60% acid solution.

The area of a rectangular rug is given by the trinomial r squared minus 5 r minus 14. What are the possible dimensions of the​ rug? Use factoring.

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Part 1
The length and the width of the rectangle are
  
enter your response here.
​(Use a comma to separate answers as​ needed.)

To find the possible dimensions of the rug, we need to factor the trinomial \(r^2 - 5r - 14\).

The factored form of the trinomial is \((r - 7)(r + 2)\).

Therefore, the possible dimensions of the rug are 7 by -2 or -7 by 2 (the negative values indicate that one of the dimensions can be negative, which may not make sense in a real-life scenario, but mathematically they are still valid factors). The length can be 7 or -7 and the width can be 2 or -2.

IS IT NEGATIVE OR POSITIVE?

I apologize for the oversight in the previous response. Let's correct that:

The factored form of the trinomial \( r^2 - 5r - 14 \) is \( (r - 7)(r + 2) \).

This means that the possible dimensions of the rug are 7 by -2. This is because the length would be \( r - 7 \) and the width would be \( r + 2 \), so the length can be 7 and the width can be -2.