If area of a kite formed by #y=f(x)=-1/2|x+6|-4# and #y=g(x)=k|x+6|-10# is #18#, find #k#?

Answer 1

#k=7/2#

#y=f(x)=-1/2|x+6|-4# represents two lines #y=-x/2-7# and #y=x/2-1# and solving them they intersect at #(-6,-4)# (to solve just add them to get #y# and then you get #x# too).
Similarly #y=g(x)=k|x+6|-10# represents two lines #y=kx+6k-10# and #y=-kx-6k-10# and solving them they intersect at #(-6,-10)#.
Assuming #k=1# we get the following graph. Observe that kite is formed with two points #(-6,-4)# and #(-6,-10)# vertically aligned and two other points formed by positively and negatively sloping pair of lines. graph{(y+x/2+7)(y-x/2+1)(y+x+16)(y-x+4)=0 [-16.54, 3.46, -11.68, -1.68]} Hence, let us consider intersection of #y=-x/2-7# and #y=kx+6k-10#. Multiplying first by #2k# we get #2ky=-kx-14k# and adding this to second we get #(2k+1)y=-8k-10# and #y=-(8k+10)/(2k+1)# and #x=-2(-(8k+10)/(2k+1)+7)=(16k+20-28k-14)/(2k+1)=(-12k+6)/(2k+1)#.
Hence, coordinates of third point are #((-12k+6)/(2k+1),-(8k+10)/(2k+1))# and area of kite is double the area of triangle formed by this point with #(-6,-4)# and #(-6,-10)#.

Area of triangle is

#1/2|(-6(-10+(8k+10)/(2k+1))+6(-4+(8k+10)/(2k+1))-(-12k+6)/(2k+1)(-4+10))|#
= #1/2|(60-(48k+60)/(2k+1)-24+(48k+60)/(2k+1)-(-72k+36)/(2k+1))|#
= #1/2|(36-(-72k+36)/(2k+1))|#
= #1/2|((72k+36+72k-36)/(2k+1))|#
= #1/2|(144k)/(2k+1)|#
And area of kite is #(144k)/(2k+1)#
As #(144k)/(2k+1)=18#
#2k+1=144/18=8# and #k=7/2#

and kite appears as

graph{(y+x/2+7)(y-x/2+1)(y+7/2x+31)(y-7/2x-11)=0 [-16.5, 3.5, -12.2, -2.2]}

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the value of ( k ), we first need to find the intersection points of the two functions ( f(x) ) and ( g(x) ). Then, we calculate the area of the kite formed by these functions and set it equal to 18. Finally, we solve for ( k ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7