The stadium has a form of a rectangle with two semicircles attached to the short sides. The long side of a rectangle is 2 times
longer than the short one. Construct the expression for the area of the stadium in terms of x, where x denotes a shorter side of the rectangle
1 answer:
Answer:
Step-by-step explanation:
The area of a rectangle can be calculated with this formula:
Where "l" is the lenght and "w" is the width.
The area of a semi-circle can be found with this formula:
Where "r" is the radius.
Let be "x" the shorter side of the rectangle (which is also the diameter of the semi-circle).
Based on the information given in the exercise, you know that:
Since the radius is half the diameter:
Observe the figure attached, which shows the stadium.
The area of the stadium is the sum of the areas of the semi-cirlcles and the rectangle.
Therefore, you can construct the following expression for the area of the stadium is terms of "x":
Simplifying it, you get:
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Explanation:
Problems 1, 2, and 5 are exponential functions of the form where b is the base of the exponent and 'a' is the starting term (when x=0).
If 0 < b < 1, then the exponential function decreases or decays. Perhaps a classic example would be to study how a certain element decays into something else. The exponential curve goes downhill when moving to the right.
If b > 1, then we have exponential growth or increase. Population models could be one example; though keep in mind that there is a carrying capacity at some point. The exponential curve goes uphill when moving to the right.
In problems 1 and 5, we have b = 2 and b = 1.1 respectively. We can see b > 1 leads to exponential growth. I recommend making either a graph or table of values to see what's going on.
Meanwhile, problem 2 has b = 0.8 to represent exponential decay of 20%. It loses 20% of its value each time x increases by 1.
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Problems 3 and 4 are linear functions of the form y = mx+b
m = slope
b = y intercept
This b value is not to be confused with the previously mentioned b value used with exponential functions. They're two different things. Unfortunately letters tend to get reused.
If m is positive, then the linear function is said to be increasing. The line goes uphill when moving to the right.
On the other hand if m is negative, then we go downhill while moving to the right. This line is decreasing.
Problem 3 has a negative slope, so it is decreasing. Problem 4 has a positive slope which is increasing.