I would like help with the question. It's part of my assigment, but I have trouble with this type of math. If someone understand
s, can you explain it to me as well? For future reference?
In Panama City in January, high tide was at midnight. The water level at high tide was 9 feet and 1 foot at low tide. Assuming the next high tide is exactly 12 hours later and that the height of the water can be modeled by a cosine curve, find an equation for water level in January for Panama City as a function of time (t).,
In Panama City in January, high tide was at midnight. The water level at high tide was 9 feet and 1 foot at low tide. Assuming the next high tide is exactly 12 hours later and that the height of the water can be modeled by a cosine curve, find an equation for water level in January for Panama City as a function of time (t).,
1 answer:
The cosine function cos(x), as you know, has a peak at x=0, a minimum at x=π, and another peak at x=2π. That is, its period is 2π. Its amplitude is 1, meaning the peak is +1 and the minimum is -1.
Problems where sine or cosine functions are used to model periodic behavior are problems in scaling. You need to match the period and amplitude of your scaled cosine function to the period and amplitude of the phenomenon you are modeling.
Here, high tides are 12 hours apart, so we need to scale x by a factor that turns 12 hours into 2π. That might be x ⇒ 2πx/12 or (π/6)x.
The high tide is 9 ft, and the low tide is 1 ft, so we need to do vertical offset and scaling to make the peak of our transformed cosine function be 9 and its minimum be 1. That difference is 8, so has an amplitude of ±4 around a midline of (9+1)/2 = 5.
Then our tide model is
.. water level = 5 +4*cos((π/6)t)
Problems where sine or cosine functions are used to model periodic behavior are problems in scaling. You need to match the period and amplitude of your scaled cosine function to the period and amplitude of the phenomenon you are modeling.
Here, high tides are 12 hours apart, so we need to scale x by a factor that turns 12 hours into 2π. That might be x ⇒ 2πx/12 or (π/6)x.
The high tide is 9 ft, and the low tide is 1 ft, so we need to do vertical offset and scaling to make the peak of our transformed cosine function be 9 and its minimum be 1. That difference is 8, so has an amplitude of ±4 around a midline of (9+1)/2 = 5.
Then our tide model is
.. water level = 5 +4*cos((π/6)t)
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1)Define the variables you will use in your model.
Let
x------->total pounds of the fruit
y-------> the total number of granola bars
TC----> total cost
Tf------> total cost of the fruit
Tg----> total cost of the granola bars
2) Write an inequality representing the total cost of your purchase. (3 points)
a. Each pound of fruit costs $4. Write an expression that shows the total cost of the fruit. Use the variable you identified in question 1
Tf=4*x
b. Each granola bar costs $1. Write an expression that shows the total cost of the granola bars. Use the variable you identified in question 1.
Tg=1*y--------> Tg=y
c. Combine the expressions from parts a and b to write an expression for the total cost. Then use this expression to write an inequality that compares the total cost with the amount you have to spend.
TC=Tf+Tg
TC=4*x+y
inequality------->
3. Write the inequality that models the number of granola bars you need to buy.
You need at least 10 granola bars
so
4. Describe in words what each of your inequalities means
inequality 1
the total cost of fruits plus the total cost of granola bars must be less than or equal to 48 dollars
inequality 2
5. Graph your system of inequalities.
using a graph tool
see the attached figure
6. Identify one point on the graph that represents a viable solution to the problem, and then identify one point that does not represent a viable solution.
Let
A-------> one point on the graph that represents a viable solution
B-------> one point on the graph that does not represent a viable solution
A(4,20)
x=4-------> total pounds of the fruit
y=20------> the total number of granola bars
check point A
substitute the values of x and y in the inequality 1
-------> is ok
substitute the values of x and y in the inequality 2
--------> is ok
the point A represents a viable solution, because satisfies both inequalities
B(10,30)
x=10-------> total pounds of the fruit
y=30------> the total number of granola bars
check point B
substitute the values of x and y in the inequality 1
-------> is not a viable solution
substitute the values of x and y in the inequality 2
--------> is ok
the point B is not a viable solution because does not satisfy the inequality 1
