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<p>The diagram shows a hemispherical bowl, of inside radius \(15\text{ cm}\), fixed so its circular rim is horizontal. When the depth of water in the bowl is \(h\text{ cm}\), the volume, \(V\text{ cm}^3\), of water in the bowl is given by \(V=\dfrac{1}{3}\pi h^2(45-h).\)</p>
<p>The bowl is initially empty, and water is poured into the bowl at the constant rate of \(10\pi\text{ cm}^3\) per second.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-4-item-0" data-list-item-path="imported-item-block-4-item-0"><div class="list-item-paragraph">Find the time taken to fill the bowl.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="imported-item-block-4-item-1" data-list-item-path="imported-item-block-4-item-1"><div class="list-item-paragraph">Find the exact rate at which the depth of water is increasing when the depth of water is \(12\text{ cm}\).<span class="inline-marks">[4]</span></div><div class="list-item-after-content">The bowl is emptied, and water is poured into the bowl at the variable rate of \(10\pi t\text{ cm}^3\) per second, where&nbsp;\(t\) seconds is the time from when pouring started.</div></li><li data-list-item-id="imported-item-block-4-item-2" data-list-item-path="imported-item-block-4-item-2"><div class="list-item-paragraph">Find the time taken to pour \(972\pi\text{ cm}^3\) of water into the bowl.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="imported-item-block-4-item-3" data-list-item-path="imported-item-block-4-item-3"><div class="list-item-paragraph">Find the rate at which the depth of water is increasing when the volume of water in the bowl is \(972\pi\text{ cm}^3\).<span class="inline-marks">[4]</span></div></li></ol>

The diagram shows a hemispherical bowl, of inside radius \(15\text{ cm}\), fixed so its circular rim is horizontal. When the depth of water in the bowl is \(h\text{ cm}\), the volume, \(V\text{ cm}^3\), of water in the bowl is given by \(V=\dfrac{1}{3}\pi h^2(45-h).\)

The bowl is initially empty, and water is poured into the bowl at the constant rate of \(10\pi\text{ cm}^3\) per second.

The bowl is emptied, and water is poured into the bowl at the variable rate of \(10\pi t\text{ cm}^3\) per second, where \(t\) seconds is the time from when pouring started.

Find the exact rate at which the depth of water is increasing when the depth of water is \(12\text{ cm}\).

Find the time taken to pour \(972\pi\text{ cm}^3\) of water into the bowl.

Find the rate at which the depth of water is increasing when the volume of water in the bowl is \(972\pi\text{ cm}^3\).