A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm long and the cross-sectional areas, in square centimeters, are 0, 19, 58, 78, 94, 105, 117, 129, 62, 40, and 0. Use the Midpoint Rule with n = 5 to estimate the volume V of the liver. V = cm3

Step-by-step explanation:

Here we have

Space in cm          Area in square centimeters  -

As we can see, and we need to divide it into 5 subintervals, where

is the length of each interval. Therefore, the subintervals are

The midpoint of the first interval is calculated as

Similarly, we obtain that the midpoints of other subintervals are

By the Midpoint rule, we have

From the given data above, we have

Therefore,

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.

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