What is the area of the shaded section?
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From the setup of the problem, the "length of the top of the bookcase, measured along the attic ceiling" will be the hypotenuse of a right triangle, the length "AB". We have both the angle between AB and AC and the length of AC (3.24 meters), so we can use trigonometric identities.
The cosine of the 40 degree angle between AB and AC is equivalent to the length of AC divided by the length of AB. Equivalently, we have:
where "h", the hypotenuse, is the length we want. Rearranging the formula to solve for h we have that
which is 4.2295... meters. Converting to centimeters (multiplying by 100) we have that h = 422.95... centimeters, or if we round the value, h = 423 centimeters.
For this case we must solve the following equation:
We apply distributive property on the right side of the equation:
We subtract 6y on both sides of the equation:
We subtract 6 from both sides of the equation:
Dividing by 6 on both sides of the equation:
So, the result is
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