[填空题]
Lohan receives an antiq 2eyd)n) dayn 5 .t3j7tej8-wgyjfa 4xue vase as a gift from her grandparents. She decides k( x c3taevl,uhp2e+ )vb9dma7(/jek to model the shape of xa )39eda ( , +buvkkjev(c/l ephm72tthe vase to calculate its volume.
She places the vase horizontally on a piece of paper and uses a pencil to mark five representative points (0,5),(7.5,10),(17,7.5),(25,3) and (35,7.5) , as shown below. These points are connected to form a symmetrical cross-section about the x axis. All units are in centimetres.
Lohan initially uses a straight line to model the section from (25,3) to (35,7.5) .
1. Determine the equation of the line that passes through these two points.
y=ax-b;a= ,b= .
Lohan thinks that a quadratic curve might be a good model for the section from (0,5) to (25,3) . She carries out a least squares regression using this model for the points she has recorded.
2. 1. Determine the equation of the least squares regression quadratic curve found by Lohan.
2. By considering the gradient of the curve at the point where x=10 , determine whether the quadratic regression curve is a good model or not.
Lohan decides that a cubic curve for the entire section from (0,5) to (35,7.5) would be a better fit.
3. Find the equation of the cubic model.
Using this model, Lohan estimates the volume of the vase by calculating the volume of revolution about the x -axis.
4. Find the volume of the vase estimated by Lohan.
Lohan subsequently fills the vase with water and discovers that the true volume is 5500 $\mathrm{~cm}^{\wedge} $3 .
5. Calculate the percentage error in Lohan's estimate of the volume.
Using the formula for percentage error, we have
ϵ= %
