The mean life of a particular brand of light bulb is 1200 hours and the standard deviation is 50 hours. we can conclude that at least 75% of this brand of bulbs will last between
(a) 1100 and 1300 hours
(b) 1150 and 1250 hours
(c) 1050 and 1350 hours
(d) 1000 and 1400 hours
(e) 950 and 1450 hours

(A) 1100 and 1300 hours

Step-by-step explanation:

In a normal distribution, we can say that 68% of the values is between the range [µ-σ;µ+σ]  with µ = the mean and σ is the standard deviation.

95% of the values are between the range [µ -2σ; µ+2σ] = [1100;1300]

99.7% of the values are between the range [µ -3σ; µ+3σ] = [1050;1350]

To find 75% of the values, we have to use the z-score

for 75% the Z-score = 1.15

This gives the range: [µ -Zσ; µ+Zσ]  ⇒ [1200 - 1.15*50;1200+1.15*50] = [1142.5 ; 1257.5]

We can say that at least 75% (or more) is in the range [1100;1300]

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12