Hello, Sarah!

1) The arm of a boom crane is 12m long.

Because of the location of the construction site, the angle of inclination

of the boom of the crane has a minimum value of 30° and a maximum of 45°.

Find the vertical displacement of the end of the boom as

a) an exact value

b) an approximate value, to the nearest tenth of a metre. Code:

*
* |
12 * |
* | y1
* |
* 30° |
* - - - - - - - - *

You're expected to the ratio of the sides of a 30-60 right triangle.

The side opposite the 30°-angle is always one-half the hypotenuse.

Hence: .$\displaystyle (b)\;y_1 \,=\,6$ m.

Code:

*
* |
12 * |
* | y2
* |
* 45° |
* - - - - - *

You should know that sides of a 45-45 right triangle are in the ratio: .$\displaystyle 1:1:\sqrt{2}$

Hence: .$\displaystyle y_2\:=\:\frac{12}{\sqrt{2}}\:=\:6\sqrt{2}$

Therefore: .$\displaystyle (a)\;y_2 - y_1 \;=\;6\sqrt{2} - 6 \;=\;6(1-\sqrt{2})\;m $

. . . . . . . . $\displaystyle (b)\;y_2-y_1 \;=\;2.485281376 \;\approx\; 2.5\;m$

.