EXAMPLE 10 Show that there is a root of the equation 4x3 − 7x2 + 3x − 2 = 0 between 1 and 2. SOLUTION Let f(x) = 4x3 − 7x2 + 3x − 2 = 0. We are looking for a solution of the given equation, that is, a number c between 1 and 2 such that f(c) = . Therefore we take a = , b = , and N = in the Intermediate Value Theorem. We have f(1) = 4 − 7 + 3 − 2 = −2 < 0 and f(2) = 32 − 28 + 6 − 2 = 8 > 0. Thus f(1) < 0 < f(2); that is N = 0 is a number between f(1) and f(2). Now f is continuous since it is a polynomial, so that the Intermediate Value Theorem says there is a number c between and such that f(c) = . In other words, the equation 4x3 − 7x2 + 3x − 2 = 0 has at least one root c in the open interval (1,2) . In fact, we can locate a root more precisely by using the Intermediate Value Theorem again. Since f(1.4) = −0.544 < 0 and f(1.5) = 0.250 > 0 a root must lie between (smaller) and (larger). A calculator gives, by trial and error, f(1.47) = −0.010208 < 0 and f(1.48) = 0.074368 > 0. So a root lies in the open interval (1.47,148)

tell you what. i'll find the length of the medians and you can fill in the blanks where they belong.

br

b = (2a,0)

r = (a/2,b/2) remember these things are medians. they go to the 1/2 way point of the line opposite the vertex they face.

br^2 =   (2a - a/2)^2 + (0 - b/2)^2

br^2 = (3/2 a) ^2 + b^2 / 4

br^2 = 9/4 a^2 + b^2 / 4   we need to find some relationship between a and b.

let's try ab = bc  

ab = 2a

bc = (2a - a)^2 + (b - 0)^2

bc = sqrt(a^2 + b^2)

ab = bc

2a = sqrt(a^2 + b^2) square both sides.

4a^2 = a^2 + b^2 subtract a^2 from both sides.

sqrt(3a^2) = sqrt(b^2)

sqrt(3)a = b

let's put b into br^2

br^2 = 9/4 a^2 + 3b^2 / 4

br^2 = 12 a^2 / 4

br^2 = 3a^2 

br = sqrt(3) * a

cp

c = (a,b) ; p = (a,0) this is another application of the distance formula, and it is a good one.

cp^2 = (a - a)^2 + (b - 0)^2

cp^2 = 0 + b^2

cp^2 = b^2 which you can see from the diagram. 

cp = sqrt(b^2)

cp = b but b = sqrt(3) * a 

cp = sqrt(3)*a

aq

a = (0,0)

q = (3/2) a, b/2

aq^2 = (3a/2 -0)^2 + (b/2 - 0)^2

aq^2 = 9a^2/4 + b^2/4

but b = sqrt(3) * a 

aq^2 = 9a^2 / 4 + (sqrt(3)*a)^2 /4

aq^2 = 9 a^2/ 4 + 3 a^2 / 4

aq^2 = 12 a^2/4

aq^2 = 3 a^2

aq = (sqrt  3) * a which agrees with the other two.

6 adults and 9 children went.

a + c = 15

12a + 9c = 153

12(6) + 9(9) = 153

72 + 81 = 153

the answer is:

the correct option is the option b.

to calculate the surface area of the right cylinder, we need to use the following formula:

from the picture we know that:

so, substituting and calculting we have:

hence, we have that the correct option is the option b.

have a nice day!