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2 answers:
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Answer:
9.207
Step-by-step explanation:
To get the amount of starbucks for the population we can do
which gives us 211.77 total Starbucks in 23 miles.
To get starbucks per mile, we divide this number by 23.
This gives us 9.207
Answer:
( 1/√ 2 , 1/√ 2 ) , ( 1/√ 2 , - 1/√ 2 ), ( -1/√ 2 , 1/√ 2 ) , ( -1/√ 2 , - 1/√ 2 )
y + 1 = - ( x + 2 ) + √ 2 , y + 1 = - ( x + 2 ) - √ 2 , y + 1 = ( x + 2 ) - √ 2
y + 1 = ( x + 2 ) + √ 2 , ( x + 2 )^2 + ( y + 1)^2 = 1
Step-by-step explanation:
Given:
- Four functions to construct a diamond:
y = −x+ √ 2, y = −x− √ 2, y = x+ √ 2, and y = x − √ 2.
Find:
a)Show that the unit circle sits inside this diamond tangentially; i.e. show that the unit circle intersects each of the four lines exactly once.
b)Find the intersection points between the unit circle and each of the four lines.
(c) Construct a diamond shaped region in which the circle of radius 1 centered at (−2, − 1) sits tangentially. Use the techniques of this section to help.
Solution:
- For first part see the attachment.
- The equation of the unit circle is given as follows:
x^2 + y^2 = 1
- To determine points of intersection we have to solve each given function of y with unit circle equation for set of points of intersection:
For: y = −x+ √ 2 , x - √ 2
And: x^2 + y^2 = 1
x^2 + (+/- * (x - √ 2))^2 = 1
x^2 + (x - √ 2)^2 = 1
2x^2 -2√ 2*x + 2 = 1
2x^2 -2√ 2*x + 1 = 0
2[ x^2 - √ 2] + 1 = 0
Complete sqr: (1 - 1/√ 2)^2 = 0
x = 1/√ 2 , x = 1/√ 2
y = -1/√ 2 + √ 2 = 1/√ 2
y = 1/√ 2 - √ 2 = - 1/√ 2
Points are: ( 1/√ 2 , 1/√ 2 ) , ( 1/√ 2 , - 1/√ 2 )
- Using vertical symmetry of unit circle we can also evaluate other intersection points by intuition:
x = - 1/√ 2
y = 1/√ 2 , -1/√ 2
Points are: ( -1/√ 2 , 1/√ 2 ) , ( -1/√ 2 , - 1/√ 2 )
- To determine the function for the rhombus region that would be tangential to unit circle with center at ( - 2 , - 1 ):
- To shift our unit circle from origin to ( - 2 , - 1 ) i.e two units left and 1 unit down.
- For shifts we use the following substitutions:
x = x + 2 ....... 2 units of left shift
y = y + 1 .......... 1 unit of down shift
- Now substitute the above shifting expression in all for functions we have:
y = −x+ √ 2 -----> y + 1 = - ( x + 2 ) + √ 2
y = −x− √ 2 -----> y + 1 = - ( x + 2 ) - √ 2
y = x- √ 2 -------> y + 1 = ( x + 2 ) - √ 2
y = x+ √ 2 ------> y + 1 = ( x + 2 ) + √ 2
x^2 + y^2 = 1 -----> ( x + 2 )^2 + ( y + 1)^2 = 1
- The following diamond shape graph would have the 4 functions as:
y + 1 = - ( x + 2 ) + √ 2 , y + 1 = - ( x + 2 ) - √ 2 , y + 1 = ( x + 2 ) - √ 2
y + 1 = ( x + 2 ) + √ 2 , ( x + 2 )^2 + ( y + 1)^2 = 1
- See attachment for the new sketch.
