Use the exponential growth function form to model the volume of dirt shoveled after x weekend.
A
a
9168ft^3
B
b
Function: y=9573-405(2)^x
5 weekends
Practice makes perfect
a
To find the volume of the trench, we will first find the area of the trench. To do so, we will find the area of the inner and the outer rectangles. The width of the trench is 4ft, so we need to subtract 8 from the length and width of outer rectangle.
Then the inner rectangle has a length of 202ft and a width of 172ft. The area of the outer rectangle minus the area of the inner rectangle gives us the area of the trench.
We can find the volume by multiplying this number by 3, the depth of the trench.
V=3056 * 3 ⇔ V=9168
The volume of the trench is 9168ft^3.
b
We know that the number of diggers doubles every weekend since each digger calls one more digger for the next weekend. Hence, the growth factor is 2. Since one person can dig 405ft^3 of dirt per weekend, a is equal to 405. We can write the function below.
y= ab^x ⇒ y= 405(2)^x
This function models the volume of dirt shoveled after x weekends. We need subtract 405 because we do not remove dirt at time zero.
y=405(2)^x ⇒ y=405(2)^x-405To write a function modeling the volume of dirt remaining after x weekends, we will use the function above and the volume of the trench.
The function y models the volume of dirt remaining after x weekends.
y=9573-405(2)^x
To find when they complete the trench, we will substitute 0 for y.
We know that 24 is between 16 and 32, which are the 4^\text{th} and 5^\text{th} powers of 2.
16 < 24 < 32 ⇔ 2^4 < 2^x < 2^5
Therefore, x is between 4 and 5.
2^4 < 2^x < 2^5 ⇔ 4 < x < 5
We can say that after the fourth weekend, there will still be some volume of dirt to be shoveled. Therefore, they complete the trench on the fifth weekend. It will take 5 weekends to complete it.
Alternative Solution
Using a calculator
To get a more accurate value, we can solve the equation by graphing.
After we substitute 0 for y into the function, we get the equation below.
0=9573-405(2)^x ⇔ 405(2)^x=9573
Now we need to create two functions. Each side of the given equation will become its own equation.
405(2)^x= 9573 ⇒ ly= 405(2)^x y= 9573
Now we want to draw these graphs using a calculator. We can do this by pressing the Y= button and typing the equations in the two first rows.
Before we graph these functions, notice that the exponential function will grow rapidly and the other function is a big number. We will not be able to see the intersection of the lines in the standard viewing window. That's why, we should change the viewing window. We can do this by pushing WINDOW.
There is one point of intersection of these lines. To find this point we can use the intersect option, which we get by pushing 2nd and TRACE. Now, we select both graphs and provide the calculator with a guess as to where the intersection might be.
The solution to the equation is x ≈ 4.56. We need to round it to integer.
x ≈ 4.56 ⇒ x ≈ 5
It will take approximately 5 weekend to complete the trench.
