Question

(PLEASE HELP WITH THIS!!!! mwa)Two cost models are proposed for producing a particular product in order to maximize a profit. As the number of units of the product are produced, the costs decrease. The cost, in dollars, for model A is represented by the function y = −0.5x2 + 125, and model B is represented by the function y = 250(0.95)x−1, where x is the number of units produced. The table gives the costs for both models in producing the first 7 units of the product.

Units Produced Model A Cost Model B Cost
2 123.00 237.50
3 120.50 225.63
4 117.00 214.34
5 112.50 203.63
6 107.00 193.45
7 100.50 183.70
If the cost for both models continues in this pattern, will the cost of model B ever be lower than the costs of model A? Explain.

a
No, the function for model A is a quadratic function that decreases at a faster rate than the function for model B, which is a decreasing exponential function.

b
No, the function for model B is a quadratic function that decreases at a slower rate than the function for model A, which is a decreasing exponential function.

c
Yes, the function for model A is an exponential function that decreases at a faster rate than the function for model B, which is a decreasing quadratic function.

d
Yes, the function for model B is an exponential function that decreases at a slower rate than the function for model A, which is a decreasing quadratic function.

Answer 1

To determine whether the cost of model B will ever be lower than the costs of model A, we can compare the cost patterns of both models as the number of units produced increases.

Looking at the given table, we can see that for each corresponding number of units produced, the cost of model B is consistently lower than the cost of model A. This pattern indicates that as the number of units produced increases, model B remains less expensive than model A.

Furthermore, we can analyze the cost functions for both models:

- Model A: y = -0.5x^2 + 125 (quadratic function)

- Model B: y = 250(0.95)^x-1 (decreasing exponential function)

From the cost functions, we can observe that the quadratic function for model A decreases at a faster rate as x increases, while the decreasing exponential function for model B decreases at a slower rate. This implies that model A's cost decreases more rapidly than model B's cost.

Based on the given information and the characteristics of the cost functions, we can conclude that the cost of model B will never be lower than the costs of model A. Therefore, the correct answer is:

b) No, the function for model B is a quadratic function that decreases at a slower rate than the function for model A, which is a decreasing exponential function.

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Answer 2

The correct answer is (a) No, the function for model A is a quadratic function that decreases faster than the function for model B, which is a decreasing exponential function.

To determine if model B will ever be lower than model A, we compare the decrease rates for both models.

Model A is represented by the quadratic function y = -0.5x^2 + 125, where x is the number of units produced. As x increases, the quadratic function decreases, but at a slower rate. This means that the cost decreases, but the rate slows down over time.

Model B is represented by the exponential function y = 250(0.95)^x, where x is the number of units produced. As x increases, the exponential function decreases faster. This means that the cost decreases, and the rate accelerates over time.

Looking at the costs given for the first 7 units produced, we can see that model A costs more initially than model B. However, as the number of units increases, the cost of Model A decreases at a slower rate than Model B.

Based on the patterns observed and the nature of the functions representing both models, the cost of model B will not be lower than model A. Model A, being a quadratic function, will eventually reach a point where the decrease in cost becomes minimal. In contrast, the exponential function of model B will continue to decrease faster. Therefore, option (a) is the correct answer.