S=6x^2 is the surface area of a cube. Find the exact change in surface area when x is increased from 6 cm to 6.1 cm .
Please Help.
It makes no sense.
I assume x is a side length.
dS/dx=12x
dS=12x dx
find dS, you are given x as 6cm, dx as .1
S(6)=6*36
S(6.1)=6*37.21 EXACT change 6*1.21cm^2=7.38
Now, using linear approximation.
dS=12*6*.1=72*.1=7.2
Now notice the exact change is 7.38, the incremental change is 7.2
This indicates a bias in an increasing function. If you had used the midpoint value initially (6.05) dx=.1
dS=12*6.05*.1=7.26
So why is it off slightly? The area equation is NOT linear, yet we used the linear approximation model (origvalue+ slope*changex)
To find the exact change in surface area when x is increased from 6 cm to 6.1 cm, we need to calculate the difference between the surface areas at these two values.
Given:
Surface area of a cube, S = 6x^2
Step 1: Calculate the surface area when x is 6 cm.
Substitute x = 6 into the equation S = 6x^2:
S1 = 6(6)^2 = 6(36) = 216 cm^2
Step 2: Calculate the surface area when x is 6.1 cm.
Substitute x = 6.1 into the equation S = 6x^2:
S2 = 6(6.1)^2 = 6(37.21) = 223.26 cm^2
Step 3: Calculate the change in surface area.
Subtract the surface area at x = 6 from the surface area at x = 6.1:
Change in surface area = S2 - S1
= 223.26 cm^2 - 216 cm^2
≈ 7.26 cm^2
Therefore, the exact change in surface area when x is increased from 6 cm to 6.1 cm is approximately 7.26 cm^2.
To find the change in surface area when x is increased from 6 cm to 6.1 cm, you can follow these steps:
1. Start with the equation for the surface area of a cube: S = 6x^2.
2. Substitute the initial value of x, which is 6 cm, into the equation to find the initial surface area:
S1 = 6(6^2) = 6(36) = 216 square cm.
3. Substitute the new value of x, which is 6.1 cm, into the equation to find the new surface area:
S2 = 6(6.1^2) = 6(37.21) = 223.26 square cm.
4. Calculate the change in surface area by subtracting the initial surface area from the new surface area:
Change in surface area = S2 - S1 = 223.26 - 216 = 7.26 square cm.
So, the exact change in surface area when x is increased from 6 cm to 6.1 cm is 7.26 square cm.