Starkist tuna company will package tuna fish in 2-L cans. Find the radius and the height of the cans if the cans have a surface area which is less that 980 cm^2.
I have no idea how to go about this, it seems like geometry to me. Can you give me some help? Please.
Sarea=PI*diameter*height + 2PI*(diameter/2)^2
Volume=PI(dia/2)^2 h
put numbers in for volume, Sarea, then solve for height in the volume equation, put that in for height in the Area equation. Solve for diameter.
volume of can = pi(r^2)(h)
2000 = pi(r^2)(h)
Surface area of the can = 2 circles + 1 rectangle, where the length of the rectangle is the circumference of the circle and its height is the height of the can.
so
2pi(r^2) + 2rpi < 980
This is actually a typical Calculus problem, where the question would ask for the dimensions of the can with the minimum surface area to hold 2 L
The way it was worded, there would be an infinite number of solutions.
Is the volume 2L? How can I solve for diameter if I don't have the height?
How do I get the volume?
You solve for diameter after putting in for height (area/2pi(dia/2)^2 into the Area equation.
It is called substitution, having two equations and solving it by substitution.
Zach. The volume was given as 2dm^3 (2liters). Use the 2dm^3, that is essential.
If you want the surface area to be equal to 980 and the volume equal to 2 L or 2000 cm^3
then you end up solving a nasty cubic
pir^3 - 980r + 4000 = 0
I found this handy cubic equation solver
http://www.1728.com/cubic.htm
and obtained two possible solutions:
r = 15.08 or r = 4.34
both give us a volume of 2 L and a surface area of 980 cm^2
So if you want your surface area to be < 980, then any r between 4.34 and 15.08 and its corresponding h = 2000/(pir^2) would give a volume of 2 L and a surface area less than 980 cm^2
I tried to solve for H in the volume equation. so, 2=3.14(r^2)H
divide 2 by 3.14=.64
.64=r^2H
.64/r^2=H
Not sure where to go from here
Of course! I can help you with this geometry problem. To find the radius and height of the cans, we'll first need to use the formula for the surface area of a cylinder.
The formula for the surface area (SA) of a cylinder is given by:
SA = 2πr² + 2πrh,
where r represents the radius and h represents the height of the cylinder.
In this case, we're given that the surface area of each can is less than 980 cm². So we can write the inequality as:
2πr² + 2πrh < 980.
Now, let's solve for either the radius (r) or the height (h) in terms of the other variable, and then substitute it into the inequality to find the range of values.
First, let's solve for the radius (r) in terms of the height (h):
2πr² + 2πrh < 980,
2πr( r + h ) < 980,
r( r + h ) < 490.
Now, let's solve for the height (h) in terms of the radius (r):
2πr² + 2πrh < 980,
2πrh < 980 - 2πr²,
h < (980 - 2πr²) / (2πr).
At this point, we have expressions for both r and h in terms of each other. However, we can't solve for exact values since we only have an inequality.
To simplify the calculations, we can use numerical methods. We can try different values for either r or h within a reasonable range and check if they satisfy the inequality. We'll use π as approximately 3.14 for the calculations.
For example, let's assume the height (h) ranges from 1 to 10 cm (you can adjust this range based on your assumption of a reasonable height for the can). We can substitute these values into the inequality and find the corresponding range of radius (r) that satisfies the inequality.
Using substitution, we have:
r( r + h ) < 490.
Let's take h = 1 cm and find the corresponding range of r:
r( r + 1 ) < 490,
r² + r < 490,
r² + r - 490 < 0.
By solving this quadratic equation, we'll find the range of r that satisfies the inequality when h = 1 cm.
You can repeat this process for different values of h and find the corresponding range of r in each case. This will help you determine the possible combinations of radius and height for the cans, given the surface area constraint.
Please note that you can also use mathematical software or calculators to solve the quadratic equation and obtain the exact values.
I hope this explanation helps you understand the steps involved in solving this problem.