[填空题]
During a trip to a fore) p/u(zx+sl(4h3v 6 m1shvs sfuc(i dttu-*ybst to forage for mushrooms, Vivl b d2b1imz7;n:dy yem.nm 4g7iane finds a giant mushroom. She decides to model the shape of the mushroom to fb1;d.g7e m:dyn lybm4n 7 z2miind 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.
Next, Viviane models the section that passes through the points (15,15),(22,10),(23.5,6) and (24,0) with a quadratic curve.
2. 1. Use your G.D.C. to find the equation of this quadratic curve.y=$ax^b+cx+d$ a = b = c = d =
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.y ≈ a(x+b)^c+d a = b = c = d =
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.≈ $cm^3$
