The equation for the base is that of a circle, so the cross sections will have a leg of length equal to the vertical distance between its halves.
x² + y² = 16 ⇒ y = ±√(16 - x²)
⇒ length = √(16 - x²) - (-√(16 - x²)) = 2 √(16 - x²)
Cross sections with thickness ∆x have a volume of
1/2 length² ∆x = 1/2 (2 √(16 - x²))² ∆x = (32 - 2x²) ∆x
since they are isosceles triangles and so their bases and heights are equal.
Then the total volume would be (D)
Answer:
Part A) see the explanation
Part B) The initial value is 60 meters and the slope is 40 m/min (see the explanation)
Step-by-step explanation:
Part A) see the attached figure with letters to better understand the problem
we know that
If two triangles are similar then the ratio of its corresponding sides is proportional
In this problem
triangles ACD and ABE are similar by AA Similarity Theorem
so
----> equation A
Remember that the slope is equal to divide the change in the y-value by the change in the x-value
so
The slope AB is equal to
The slope AC is equal to
Rewrite the equation A
therefore
The slope of AB is equal to the slope of AC
Part B)
Let
x ---->the amount of time in minutes
y ----> the depth in meters
Find the initial value
we know that
The y-intercept or initial value is the value of y when the value of x is equal to zero
In this context the y-intercept is the depth of the shark when the time is equal to zero, so is the initial depth of the shark
Looking at the graph
For x=0
y=60 meters
therefore
The initial value is 60 meters
<em>Find the slope</em>
The formula to calculate the slope between two points is equal to
Take two points from the graph
C(0,60) and B(1,100)
substitute
That means ---->The depth of the shark increases at the rate of 40 meters per minute
