Problema Solution
The shape of a stone arch in a park can be modeled by the graph of the equation y = -x2+6x where x and y are measured in feet. On a coordinate plane, the ground is represented by the x-axis. a. make a table of values that shows the height of the stone arch for x=0,1,2,3,4 and 5 feet. b. plot the ordered pairs in the table from part (a) as points in a coordinate plane. Connect the points with a smooth curve. c. how wide is the base of the arch? justify your answer using the zeros of the given funtion. d. at how many points does the arch reach a height of 9 feet: justify your answer algebraically.
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STEP1
GIVEN INFORMATION
The shape of a stone arch in a park can be modeled by the graph of the equation y = -x2+6x where x and y are measured in feet.
STEP2
a) TO KNOW HEIGHT of stone arch for x= 0 ,1, 2,3 ,4 5 feet
simply put values of x in equation y= - x^2 +6x
here is table for differnt x values
x y (height)
0 0
1 5
2 8
3 9
4 8
5 5
step3
b)
orderd pairs (x,y) are
(0 ,0)
(1,5)
(2,8)
(3,9)
(4,8)
(5 ,5)
step4
c)
to know how wide is base of arch we require to solve
y= -x^2 +6x =0
x( -x +6)=0
we gets
x=0
x= 6
hence base is 6 feet wide
we can also justify by zeros of function
let f(x) =-x^2 +6x
this function will have max. zero 2 as max, power 2
now at x=0 and x=6
f(0)=0 and f(6)=0
so functions zeros are at x=0 and x=6
so roots are x1=0 and x2=6
so x varation is from o to 6 feet
so base is 6 feet wide
step5
to know at how many points arch raech at 9 feet
put y=9 in y= -x^2+ 6x
-x^2 +6x =9
x^2 -6x +9=0
( x- 3)^2=0
x= 3 or -3
so points are (3,9) and (-3,9)
so at 2 points arch attains 9 feets height