Michael baked 60 oatmeal cookies and 45 chocolate chip cookies to package in plastic containers for her friends at school. She w
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Answer:
146 ft²
21.6 tiles
1.8 boxes
Step-by-step explanation:
Please find attached an image of the family room used in answering this question
the family room has the following shapes : 2 triangles and one rectangle
the area of the family room can be determined by calculating the area of each of the shape and adding the 3 areas together
area of a rectangle = length x breadth
16 x 7 = 112 ft²
Area of a triangle = 1/2 x base x height
Area of the smaller triangle = (1/2) x 4 x 3 = 6 ft²
Area of the bigger triangle = (1/2) x 8 x 7 = 28 ft²
Sum of the areas = 112 + 6 + 28 = 146 ft²
b.
1. First convert the area of the room to inches
1 ft = 12 in
146 x 12 = 1752 in²
2. the next step is to determine the area of the tile
area of a square = length²
9² = 81 in²
3. Divide the area of the room by the area of the tile
1752 / 81 = 21.6 tiles
c. total number of boxes that would be bought = 21.6 /12 = 1.8 boxes
If you know that
then it's possible to rewrite the given limit so that it resembles the one above. Then the limit itself would be some expression involving .
For starters, we have
Let . Then as
, we also have
, and
So in terms of , the limit is equivalent to
Now use some of the properties of limits: the above is the same as
The first limit is trivial; , so its value is 1. The second limit comes out to
To see why this is the case, replace , so that
as
, and
Then the limit we're talking about has a value of
# # #
Another way to do this without knowing the definition of as given above is to take apply exponentials and logarithms, but you need to know about L'Hopital's rule. In particular, write
(where the notation means , just to get everything on one line).
Recall that
if is continuous at
.
is continuous everywhere, so we have
For the remaining limit, write
Now as , both the numerator and denominator approach 0, so we can try L'Hopital's rule. If the limit exists, it's equal to
and our original limit comes out to the same value as before, .