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Monica [59]
3 years ago
10

Alabama Instruments Company has set up a production line to manufacture a new calculator. The rate of production of these calcul

ators after t weeks is
dx/dt=5000(1−100/(t+10)2)calculators/week

(Notice that production approaches 5000 per week as time goes on, but the initial production is lower because of the workers’ unfamiliarity with the new techniques.) Find the number of calculators produced from the beginning of the third week to the end of the fourth week.
Mathematics
1 answer:
umka21 [38]3 years ago
6 0

Answer:

The number of calculators is 4871

Step-by-step explanation:

If we integrate dx/dt we get x, which is the number of calculators. To find the number of calculators between the beginning of third week to the end of fourth week (the beginning of fifth week), this integration must be evaluated at t between 3 and 5.

x=\int\limits^5_3 {\frac{dx}{dt}} \, dt =\int\limits^5_3 {5000(1-\frac{100}{(t+10)^2})} \, dt

the result of the integration is:

x=5000(t+\frac{100}{t+10}) to be evaluated between 3 and 5, which is:

x=5000(5+\frac{100}{5+10})-5000(3+\frac{100}{3+10})=\frac{190000}{39}=4871.8

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Using the Factor Theorem, the polynomials are given as follows:

1. P(x) = x^5 + 2x^4 - 7x^3 + x^2

2. P(x) = 0.8(x^4 - 4x^3 - 16x^2 + 64x)

3. P(x) = -0.1(x³ - 4x² - 3x + 18)

<h3>What is the Factor Theorem?</h3>

The Factor Theorem states that a polynomial function with roots x_1, x_2, \codts, x_n is given by:

f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)

In which a is the leading coefficient.

Item a:

The parameters are:

a = 1, x_1 = x_2 = 1, x_3 = x_4 = 0, x_5 = -4

Hence the equation is:

P(x) = (x - 1)²x²(x + 4)

P(x) = (x² - 2x + 1)(x + 4)x²

P(x) = (x³ + 2x² - 7x + 1)x²

P(x) = x^5 + 2x^4 - 7x^3 + x^2

Item b:

The roots are:

x_1 = x_2 = 4, x_3 = 0, x_4 = -4

Hence:

P(x) = a(x - 4)²x(x + 4)

P(x) = a(x² - 16)x(x - 4)

P(x) = a(x³ - 16x)(x - 4)

P(x) = a(x^4 - 4x^3 - 16x^2 + 64x)

It passes through the point x = 5, P(x) = 36, hence:

45a = 36.

a = 4/5

a = 0.8

Hence:

P(x) = 0.8(x^4 - 4x^3 - 16x^2 + 64x)

Item 3:

The roots are:

x_1 = x_2 = 3, x_3 = -2

Hence:

P(x) = a(x - 3)²(x + 2)

P(x) = a(x² - 6x + 9)(x + 2)

P(x) = a(x³ - 4x² - 3x + 18)

For the y-intercept, x = 0, y = -1.8, hence:

18a = -1.8 -> a = -0.1

Thus the function is:

P(x) = -0.1(x³ - 4x² - 3x + 18)

More can be learned about the Factor Theorem at brainly.com/question/24380382

#SPJ1

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First ansewr gets Brainly!
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Hope this helps:)
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