Step 1a

The given angle of elevation = 30, Distance from the base of the tower to Sarah = 50 meters.

Let h be the height of the tower.

Using the tangent function, we have:

\$"tan"(30) = "h"/50\$

\$"h" = "tan"(30) times 50\$

h = 28.86 m

Explanation: Use the tangent function to determine tower height based on the provided angle of elevation and distance.

Step 1b

Convert the angles from degrees to radians: \$"Radians" = "Degree" times (pi)/180\$

Plug the value in the formula: \$"Radians" = 30 times (pi)/180\$

\$"Radians" = \cancel(30) times (pi)/\cancel(180)^6\$

\$"Radians" = (pi)/6\$

Therefore, the height of the tower is approximately 28.86 meters, and the angle in radians is \$π/6\$​.

Explanation: Calculate radians from degrees using the conversion formula to determine the angle in radians.

Step 2a

To find the lengths of sides XY and YZ in triangle XYZ, we can use the law of sines, which states:
\$"a"/("sin""A") = "b"/("sin""B") = "c"/("sin""C")\$

Given that angle X measures 48° and side XZ measures 16 units, we can find angle Z using the fact that the sum of angles in a triangle is 180°:
Z = 180 - ( X + Y)
Z = 180 - ( 48 + 64)
Z = 68°

Explanation: Here, introduce the law of sines formula and then find the angle Z value.

Step 2b

Now apply the law of sines to find the lengths of sides XY.

\$("XY")/("sin""X") = ("XZ")/("sin""Z")\$

\$("XY")/("sin"(48)) = 16/("sin"(68))\$

\$"XY" = (16 times "sin"(48))/("sin"(68))\$
XY = 12.81 units

Explanation: Here, use the law of sines rule for find the length of XY.

Step 2c

Now apply the law of sines to find the lengths of sides YZ.

\$("YZ")/("sin""Y") = ("XZ")/("sin""Z")\$

\$("YZ")/("sin"(64)) = 16/("sin"(68))\$

\$"YZ" = (16 times "sin"(64))/("sin"(68))\$

YZ = 15.5 units

So, the sides XY and YZ lengths are approximately 12.81 units and 15.5 units, respectively.

Explanation: Here also use the law of sines rule for find the length of YZ.