The smallest number of bulbs to be purchased so that the succession of bulbs produces light for at least 40 months with probability at least 0.9772 is 16.

Step-by-step explanation:

Solving a quadratic equation:

Given a second order polynomial expressed by the following equation:

$ax^{2} + bx + c, a\neq0$ .

This polynomial has roots $x_{1}, x_{2}$ such that $ax^{2} + bx + c = a(x - x_{1})*(x - x_{2})$ , given by the following formulas:

$x_{1} = \frac{-b + \sqrt{\Delta}}{2*a}$

$x_{2} = \frac{-b - \sqrt{\Delta}}{2*a}$

$\Delta = b^{2} - 4ac$

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

n values from a normal distribution:

The mean is $\mu n$ and the standard deviation is $s = \sigma\sqrt{n}$

A company manufactures a brand of lightbulb with a lifetime in months that is normally distributed with mean 3 and variance 1.

This means that $\mu = 3, \sigma = \sqrt{1} = 1$

For n bulbs:

The distribution for the sum of n bulds has $\mu = 3n, \sigma = \sqrt{n}$

What is the smallest number of bulbs to be purchased so that the succession of bulbs produces light for at least 40 months with probability at least 0.9772?

We want that: $S_{n} \geq 40 = 0.9772$ .

This means that when X = 40, Z has a pvalue of 1 - 0.9772 = 0.0228, that is, when $X = 40, Z = -2$ . So

$Z = \frac{X - \mu}{\sigma}$

$-2 = \frac{40 - 3n}{\sqrt{n}}$

$-2\sqrt{n} = 40 - 3n$

$3n - 2\sqrt{n} - 40 = 0$

Using $y = \sqrt{n}$

$3y^2 - 2y - 40 = 0$

Which is a quadratic equation with $a = 3, b = -2, y = -40$

$\Delta = b^{2} - 4ac = (-2)^2 - 4(3)(-40) = 484$

$y_{1} = \frac{-(-2) + \sqrt{484}}{2*3} = 4$

$y_{2} = \frac{-(-2) - \sqrt{484}}{2*3} = -...$

Since y and n have both to be positive:

$y = \sqrt{n}$

$\sqrt{n} = 4$

$(\sqrt{n})^2 = 4^2$

$n = 16$

The smallest number of bulbs to be purchased so that the succession of bulbs produces light for at least 40 months with probability at least 0.9772 is 16.

4 0

3 years ago

The perimeter of the rectangle below is 84 unit . Find the length side of XY

lesya [120]

Answer:

25 unit

Step-by-step explanation:

See attachment for the missing figure.

Assuming the complete question is:

Side UY = 4z-1

Side UV = 5z+3

Perimeter of rectangle is given by:

2UY+2UV=P

2(4z-1)+2(5z+3)=84

8z-2+10z+6=84

18z+4=84

18z = 84 -4 =>

18z = 80

z= 4.45 unit

Side UV = 5z+3 => 5(4.45) + 3

Side UV ≈ 25 unit

SInce, Side UV = Side XY

Side XY≈ 25 unit

5 0

3 years ago

I dont know how to do this

inn [45]

The dimensions of the rectangle are: length= 18 in and width =1 in.

<h3>Quadrilaterals</h3>

There are different types of quadrilaterals, for example, square, rectangle, rhombus, trapezoid, and parallelogram. Each type is defined accordingly to its length of sides and angles. For example, in a rectangle, the opposite sides are equal and parallel and their interior angles are equal to 90°.

The area of a rectangle can be found for the formula : l*w, where b = length and w =width. The question gives that the area is 18 in².

For this question, the length exceeds its width by 17 inches - l=w+17. Thus, from the value of area given, you can find the values of the length and width of the rectangle.

A=l*w

18=(w+17)*w

18=w²+17w

w²+17w-18=0

Next step will be solve the previous equation ( W²+17W-18=0)

$w_{1,\:2}=\frac{-17\pm \sqrt{17^2-4\cdot \:1\cdot \left(-18\right)}}{2\cdot \:1}\\ \\ w_{1,\:2}=\frac{-17\pm \:19}{2\cdot \:1}$

Therefore,

$w_1=\frac{-17+19}{2\cdot \:1}=1\\ \\ \\ w_2=\frac{-17-19}{2\cdot \:1}=-18$

For dimensions, only positive numbers must be used. Then, the width is equal to 1 inch.

As, the area (l*w) is 18 in², you have.

18=l*w

18=l*1

l=18 in

Read more about the area of rectangle here:

brainly.com/question/25292087

#SPJ1

3 0

2 years ago

Select the multiplication sentence that applies the associative property of multiplication to the example. Example: 1 × (9 × 5)

Firdavs [7]

The example is 1 x (9 x 5) = 45

Using the associative property of multiplication, we basically rearrange the parenthesis to group the first two terms like so: (1 x 9) x 5

So this is why choice D (1 x 9) x 5 = 45 is the answer

This means that 1 x (9x 5) = (1 x 9) x 5 is a true equation

4 0

4 years ago