State whether the equation represents exponential growth, exponential decay, or neither.
yequals4 times 1.31 Superscript x
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Part 1
Choose the correct answer below.
Exponential growth
Exponential decay
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.