(a) If logo a = 1.3010 and log\(_{10}\)b - 1.4771. find the value of ab (b) In the diagram. O is the centre of the circle,< ACB = 39\(^o\) and < CBE = 62\(^...

(a) If logo a = 1.3010 and log\(_{10}\)b - 1.4771. find the value of ab

(b)

In the diagram. O is the centre of the circle,< ACB = 39\(^o\) and < CBE = 62\(^o\). Find: (i) the interior angle AOC;

(ii) angle BAC.

10^1.3010 = a
a = 19.9526

10^1.4771 = b
b = 30.059

ab = 19.9526 * 30.059
ab = 600.001

              A
              |\
              | \
    e |  \ c
              |   \
              |    \
   -------O-----
              |    / 
              |  / d
              | /
              |/
              B

angle AOB = 180 - angle ACB
angle AOB = 180 - 39
angle AOB = 141

CD = AC = BC
CD = d
sin(62) = CD/CE
CD = CE * sin(62)

CE^2 = AC^2 + AE^2 - 2(AC)(AE)cos(39)
CE^2 = d^2 + e^2 - 2de(cos(141))

Answer Details

(a)

We know that log₁₀a = 1.3010. Therefore, we can use the definition of logarithms to write:

10^1.3010 = a
a = 19.9526

We also know that log₁₀b = 1.4771. Using the same process as above, we can find that:

10^1.4771 = b
b = 30.059

Finally, we can find the value of ab by multiplying these two values:

ab = 19.9526 * 30.059
ab = 600.001

Therefore, ab is approximately equal to 600 (to the nearest whole number).

(b)

              A
              |\
              | \
    e |  \ c
              |   \
              |    \
   -------O-----
              |    / 
              |  / d
              | /
              |/
              B

We can start by using the fact that angles in a triangle add up to 180 degrees. Therefore, angle AOB is:

angle AOB = 180 - angle ACB
angle AOB = 180 - 39
angle AOB = 141

Since O is the center of the circle, we know that AC and BC are radii of the circle. Therefore, they are equal in length. We can use this fact to find the length of CD, since angle CDE is a right angle:

CD = AC = BC
CD = d
sin(62) = CD/CE
CD = CE * sin(62)

Now we can use the cosine rule to find the length of CE:

CE^2 = AC^2 + AE^2 - 2(AC)(AE)cos(39)
CE^2 = d^2 + e^2 - 2de(cos(141))

We can also use the cosine rule to find the