<p>An architect is designing a building, \(ABCD\), using the model shown below.<br>To allow more natural light into the building, a \(12\) m glass roof, \(BC\), is installed at an angle of \(\theta\) to the vertical. Another glass roof, \(AB\), of \(7\) m, is installed such that it is perpendicular to \(BC\). The vertical height of the brick wall, \(AD\), is given as&nbsp;\(h\) m.</p>
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<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Show that&nbsp;\(h = 12 \cos \theta - 7 \sin \theta\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Express \(h\) in the form \(R \cos(\theta + \alpha)\), where&nbsp;\(R &gt; 0\) and \(0 &lt; \alpha &lt; \frac{\pi}{2}\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c"><div class="list-item-paragraph">Find the value of \(\theta\) for which \(h = 8\) m.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d"><div class="list-item-paragraph">The building owner makes a request to have the brick wall to have a height of at least \(14\) m. Is the architect able to meet the building owner's request? Explain your answer.<span class="inline-marks">[2]</span></div></li></ol>

An architect is designing a building, \(ABCD\), using the model shown below.
To allow more natural light into the building, a \(12\) m glass roof, \(BC\), is installed at an angle of \(\theta\) to the vertical. Another glass roof, \(AB\), of \(7\) m, is installed such that it is perpendicular to \(BC\). The vertical height of the brick wall, \(AD\), is given as \(h\) m.

Show that \(h = 12 \cos \theta - 7 \sin \theta\).

Express \(h\) in the form \(R \cos(\theta + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\).

Find the value of \(\theta\) for which \(h = 8\) m.

The building owner makes a request to have the brick wall to have a height of at least \(14\) m. Is the architect able to meet the building owner's request? Explain your answer.

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions