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106 = 34 +4t

18 t shirts

Step-by-step explanation:

total profit = profit from hats + profit from tshirts

profit from hats = 34

profit from tshirts = 4* t  where t is the number of tshirts sold

total profit = 34 + 4t

The total profit was 106

106 = 34 +4t

Subtract 34 from each side

106-34 = 34-34 +4t

72 = 4t

Divide each side by 4

72/4 = 4t/4

18 = t

He sold 18 t shirts

The hyperbola has the smalles distance between the vertex and the focus belonging to the same branch is 2.42:

Option B. (x-1)^2/12^2-(y-11)^2/8^2=1

Step-by-step explanation:

Form 1: (x-h)^2/a^2-(y-k)^2/b^2=1

Form 2: (y-k)^2/a^2-(x-h)^2/b^2=1

Distance between the vertex and the focus: c-a

c^2=a^2+b^2

A. (x+10)^2/11^2-(y-3)^2/10^2=1

a^2=11^2→a=11

b^2=10^2

c^2=a^2+b^2→c^2=11^2+10^2→c^2=121+100→c^2=221→sqrt(c^2)=sqrt(221)→

c=14.87

c-a=14.87-11→c-a=3.87

B. (x-1)^2/12^2-(y-11)^2/8^2=1

a^2=12^2→a=12

b^2=8^2

c^2=a^2+b^2→c^2=12^2+8^2→c^2=144+64→c^2=208→sqrt(c^2)=sqrt(208)→

c=14.42

c-a=14.42-12→c-a=2.42

C. (y-8)^2/9^2-(x+6)^2/12^2=1

a^2=9^2→a=9

b^2=12^2

c^2=a^2+b^2→c^2=9^2+12^2→c^2=81+144→c^2=225→sqrt(c^2)=sqrt(225)→

c=15

c-a=15-9→c-a=6

D. (y-14)^2/9^2-(x+28)^2/11^2=1

a^2=9^2→a=9

b^2=11^2

c^2=a^2+b^2→c^2=9^2+11^2→c^2=81+121→c^2=202→sqrt(c^2)=sqrt(202)→

c=14.21

c-a=14.21-9→c-a=5.21

The smallest distance between the vertex and the focus belonging to the same branch (c-a) is 2.42 (Option B.)