[填空题]
During a trip to a forest to fora2a-. ki a.ynsgge for mushrooms, Viviane findsxfp.c3pb k owi/* .1qq a giant mushroom. She decides to model the sfq3ic /.wq*pk xop b1.hape of the mushroom to find its volume.
After taking a photo of the mushroom and zooming in to get its real size, she rotates the photograph and estimates that the cross-section passes through the points (0,3) (15,3),(15,15),(22,10),(23.5,6) and (24,0) , where all measurements are in centimetres. The cross-section is symmetrical about the x -axis, as shown below.
Viviane models the section from (0,3) and (15,3) with a straight line.
1. Write down the equation of the line that passes through these points.
y= .
Next, Viviane models the section that passes through the points (15,15),(22,10) , (23.5,6) and (24,0) with a quadratic curve. it is ( , )
2. 1. Use your G.D.C. to find the equation of this quadratic curve.
y≈-1.82(x-a)$^2$+a;a= .
2. By considering the gradient of the curve at the point (15,15) , explain why this may not be a good model.
Viviane thinks she can obtain a better model if a quadratic passing through the point (24,0) with a maximum point at (15,15) is used.
3. Find the equation of this model, in the form y=a(x-h)^{2}+k .
Using this new model, Viviane proceeds to estimate the volume of the mushroom by finding the volume of revolution about the x -axis.
4. 1. Write down an expression for her estimate as a sum of two integrals.
2. Find the volume of the mushroom estimated by Viviane.
V≈ cm$^2$ (round number)
