Question on Circles, Finding unknown values of attributes of the circle

Circle is a geometrical figure with basic attributes diameter, radius, circumference, arc, chord etc. Below is a Geometry construction for the circle and to find unknown attributes value.

Topic : Circle and its attributes

Tangent is a straight line which just touches the circumference of the circle at a single point.

Question : Given VQ is tangent to the circle O at Q, QS is a diameter of the circle O, arc PQ = 115⁰, angle RPS = 36⁰

Find a) angle R ; b) angle S ; c) arc SR ; d) arc QR ; e) angle QPR ; f) angle QPS ; g) angle QTP











Solution :











Given : mPQ = 115⁰
‹RPS = 36⁰

a) ‹R
Since QS is the diameter,
So ‹QRS = 90⁰
(as angle in a semi circle is a right angle)

b) ‹S
As mPQ = 115⁰
So ‹PSQ = 1/2 * (115⁰) = 57.5⁰
(angle subtented by an arc at any point on the circle is half of the angle subtented by it at the center)

c) mSR
given ‹SPR = 36⁰
So mSR = 2(36) = 72⁰

d) mQR
now mQS = 180⁰
mQR = mQS - mSR
mQR = 180⁰ - 72⁰
mQR = 108⁰

e) ‹QPR
mQR = 108⁰
So ‹QPR = 1/2*(108) = 54⁰

f) ‹QPS
‹QPS = 90⁰(as QS is the diameter)

g) ‹QTP
mQP = 115⁰
so ‹QSP = 1/2 *(115⁰) = 57.5⁰
now in triangle PTS
‹QTP = 180⁰ - (‹TPS + ‹PST)
‹QTP = 180⁰ - (36⁰ + 57.5⁰)
‹QTP = 180⁰ - 93.5⁰
‹QTP = 86.5⁰

Well these are the bits of circle, to get more examples and get practiced, contact geometry help.

A Simple Word Problem on Finding an Integral of an Expression/Temperature vs Time problem

Here is a Multi choice word problem on Temperature and Time to Evaluate the temperature by Integration. Also converting the value into degrees. You can find similar set of questions at TutorVista Blogs.

Topic : Integration.

Calculation in the solution will help you to identify correct choice and get reasons for, why remaining choices are incorrect.

Question : An observer measures the outside temperature every hour from noon until midnight, recording the temperatures in the following table
Time Temp
N -----63
1 ----- 65
2 ----- 66
3 ----- 68
4 ----- 70
5 ----- 69
6 ----- 68
7 ----- 68
8 ----- 65
9 ----- 64
10 ---- 62
11 ---- 58
M ----- 55

Then the average temperature for the 12-hour period is

a. 71
b. 76
c. 65
d. 66

Solution :
Choice c is correct.

We are looking for the average value of a continuous function (temp.) for which we know values at discrete times that are one unit apart.
We need to find
Average (f) = 1/(b-a)_a∫^bf(x)dx
Without having a formula for f(x)
The integral, however can be approximated by the Trapezoidal Rule, taking h=1

T = h/2(y₀ + 2y₁ + 2y₂ + ..... + 2y₁₁ + 2y₁₂)

= 1/2 (63 + 2*65 + 2*66 + ...... + 2*58 + 2*55)

= 782

average (f)approximately 1/(b-a)T = 1/12*782 = 65.17
Therefore, the average temperature =65 degrees

Choice a is incorrect because of the error in considering the interval between noon to midnight as 13 hours instead of 12 hours.

Choice b is incorrect because of the error in considering the interval between noon to midnight as 11 hours instead of 12 hours.

Choice d is incorrect because of error in calculating the trapezoidal approximation T as 792 instead of 782.

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