Assume P(xp,yp), A(xa,ya), etc. We know that rotation rule of 90<span>° clockwise about the origin is R_-90(x,y) -> (y,-x) For example, rotating A about the origin 90</span><span>° clockwise is (xa,ya) -> (ya, -xa) or for a point at H(5,2), after rotation, H'(2,-5), etc.
To rotate about P, we need to translate the point to the origin, rotate, then translate back. The rule for translation is T_(dx,dy) (x,y) -> (x+dx, y+dy)
So with the translation set at the coordinates of P, and combining the rotation with the translations, the complete rule is: T_(xp,yp) R_(-90) T_(-xp,-yp) (x,y) -> </span>T_(xp,yp) R_(-90) (x-xp, y-yp) -> T_(xp,yp) (y-yp, -(x-xp)) -> (y-yp+xp, -x+xp+yp)
Example: rotate point A(7,3) about point P(4,2) => x=7, y=3, xp=4, yp=2 => A'(3-2+4, -7+4+2) => A'(5,-1)
The amount Cindy should increase for each dimension of the scaled model will be 12 meters.
<h3>What is the scale factor?</h3>
The scale factor is defined as the proportion of the new image's size to that of the previous image. decision-making.
Given that:-
The scaled model of the container has dimensions of 2m by 4m by 6m. Cindy has decided to increase each dimension of the scaled model by the same amount in order to produce a container with a volume of 84 times the volume of the scale model.
The scalemodel will be solved as follows:-
Present volume = 2 x 4 x 6 = 48 cubic meters
Let the lengths be increased by x meters now the new volume will be:-
( 2 + x ) ( 4 + x ) ( 6 + x ) = 48 x 84
( x² + 6x + 8 ) ( 6 + x ) = 48 x 84
( x³ + 44x + 12x² - 3984 ) = 0
By solving the cubic equation we will get the value of x = 12 meters.
Therefore the amount Cindy should increase for each dimension of the scaled model will be 12 meters.
