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Tema [17]
3 years ago
11

The lifespan of a 60-watt lightbulb produced by a company is normally distributed with a mean of 1450 hours and a standard devia

tion of 8.5 hours. if a 60-watt light bulb produced by this company is selected at random, what is the probability that its lifespan will be between 1440 and 1465 hours
Mathematics
1 answer:
Flura [38]3 years ago
8 0
To evaluate the probability that the lifespan will be between 1440 and 1465 hours will be given by:
P(1440<x<1465)
using the z-score formula we obtain:
z=(x-μ)/σ
where:
μ=1450
σ=8.5
hence
when x=1440
z=(1440-1450)/8.5
z=-1.18
P(z<-1.18)=0.1190

when x=1465
z=(1465-1450)/8.5
z=1.77
P(z<1.77)=0.9625

hence:
P(1440<x<1465)
=0.9625-0.1180
=0.8445
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Write the polynomial f(x)=x^4-10x^3+25x^2-40x+84. In factored form
Verizon [17]
<h2>Steps:</h2>

So firstly, to factor this we need to first find the potential roots of this polynomial. To find it, the equation is \pm \frac{p}{q}, with p = the factors of the constant and q = the factors of the leading coefficient. In this case:

\textsf{leading coefficient = 1, constant = 84}\\\\p=1,2,3,4,6,7,12,14,21,28,42,84\\q=1\\\\\pm \frac{1,2,3,4,6,7,12,14,21,28,42,84}{1}\\\\\textsf{Potential roots =}\pm 1, \pm 2,\pm 3,\pm 4,\pm 6, \pm 7,\pm 12,\pm 14,\pm 21,\pm 28,\pm 42,\pm 84

Next, plug in the potential roots into x of the equation until one of them ends with a result of 0:

f(1)=(1)^4-10(1)^3+25(1)^2-40(1)+84\\f(1)=1-10+25-40+84\\f(1)=60\ \textsf{Not a root}\\\\f(2)=2^4-10(2)^3+25(2)^2-40(2)+84\\f(2)=16-10*8+25*4-80+84\\f(2)=16-80+100-80+84\\f(2)=80\ \textsf{Not a root}\\\\f(3)=3^4-10(3)^3+25(3)^2-40(3)+84\\f(3)=81-10*27+25*9-120+84\\f(3)=81-270+225-120+84\\f(3)=0\ \textsf{Is a root}

Since we know that 3 is a root, this means that one of the factors is (x - 3). Now that we know one of the roots, we are going to use synthetic division to divide the polynomial. To set it up, place the root of the divisor, in this case 3 from x - 3, on the left side and the coefficients of the original polynomial on the right side as such:

  • 3 | 1 - 10 + 25 - 40 + 84
  • _________________

Firstly, drop the 1:

  • 3 | 1 - 10 + 25 - 40 + 84
  •     ↓
  • _________________
  •     1

Next, multiply 3 and 1, then add the product with -10:

  • 3 | 1 - 10 + 25 - 40 + 84
  •     ↓ + 3
  • _________________
  •     1  - 7

Next, multiply 3 and -7, then add the product with 25:

  • 3 | 1 - 10 + 25 - 40 + 84
  •     ↓ + 3  - 21
  • _________________
  •     1  - 7 + 4

Next, multiply 3 and 4, then add the product with -40:

  • 3 | 1 - 10 + 25 - 40 + 84
  •     ↓ + 3  - 21 + 12
  • _________________
  •     1  - 7  +  4  - 28

Lastly, multiply -28 and 3, then add the product with 84:

  • 3 | 1 - 10 + 25 - 40 + 84
  •     ↓ + 3  - 21 + 12  - 84
  • _________________
  •     1  - 7  +  4  - 28 + 0

Now our synthetic division is complete. Now since the degree of the original polynomial is 4, this means our quotient has a degree of 3 and follows the format ax^3+bx^2+cx+d . In this case, our quotient is x^3-7x^2+4x-28 .

So right now, our equation looks like this:

f(x)=(x-3)(x^3-7x^2+4x-28)

However, our second factor can be further simplified. For the second factor, I will be factoring by grouping. So factor x³ - 7x² and 4x - 28 separately. Make sure that they have the same quantity inside the parentheses:

f(x)=(x-3)(x^2(x-7)+4(x-7))

Now it can be rewritten as:

f(x)=(x-3)(x^2+4)(x-7)

<h2>Answer:</h2>

Since the polynomial cannot be further simplified, your answer is:

f(x)=(x-3)(x^2+4)(x-7)

6 0
3 years ago
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