Respuesta :
Answer:
ΔV = 337.5000 cm^3; ΔV/V = 0.1; ΔV/V = 10%
ΔA = 90.0000 cm^3; ΔA/A = 0.0667; ΔA/A = 6.6667 %
Explanation:
Volume is given by:
[tex]V =L^3 = 3375 cm^3[/tex]
If we derive this formula, we can estimate the maximum error on the volume:
[tex]\Delta V = 3*L^2*\Delta L[/tex]
[tex]\Delta V = 3*15^2*0.5 = 337.5000 cm^3[/tex]
[tex]\Delta V /V = 0.1[/tex]
[tex]\Delta V /V = 10 %[/tex]
The area is given by:
[tex]A = 6*L^2 = 1350 cm^2[/tex]
If we derive this formula, we can estimate the maximum error on the volume:
[tex]\Delta A = 12*L*\Delta L[/tex]
[tex]\Delta A = 12*15*0.5 = 90.0000 cm^3[/tex]
[tex]\Delta A /A= 0.0667[/tex]
[tex]\Delta A /A= 6.6667 %[/tex]
Bi definiton of volome and area of a cube, maximum possible error, relative error, and percentage error, you obtain that
- The maximum error on the volume is 337.5 cm³.
- The relative error on the volume is 0.1
- The percentage error is 10%.
- The maximum error on the surface area is 90 cm².
- The relative error on the surface area is 0.0667
- The percentage error on the volume is 6.67%.
- Volume of a cube
First, you have to know that the volume of a cube is equal to the measure of its side to the cube. That is to say:
Volume = side x side x side = side³
The derivative of a power is equal to the exponent multiplied by the base raised to the power minus one.
That is, if you have a number x raised to the power n, its derivative is equal to n multiplied by xⁿ⁻¹.
Then, deriving the volume expression, you obtain that:
[tex]\frac{dV}{dx}[/tex]=3×side²
So:
dV=3×side²×dx
The edge of a cube was found to be 15 cm with a possible error in measurement of 0.5 cm. This is,side is equal to 15 cm and dx is equal to 0.5 cm. Replacing in the previous expression you get:
dV= 3×(15 cm)²×0.5cm
dV=337.5 cm³
The maximum error on the volume is 337.5 cm³.
The relative error is obtained by dividing the maximum error by the total volume. Being, in this case, the volume:
V= (15 cm)³= 3375 cm³
The relative error is calculated as:
[tex]\frac{dV}{V} =\frac{337.5 cm^{3}}{3375cm^{3} }[/tex]
[tex]\frac{dV}{V}[/tex]= 0.1
The relative error on the volume is 0.1
Finally, the percentage error is calculated by multiplying the relative error by 100. In this case, the percentage error is calculated as:
0.1× 100 = 10%
The percentage error on the volume is 10%.
- Surface area of a cube
To calculate the area of a cube, the following formula is applied
A = 6×side×side= 6×side²
That is, the area of one of its faces is multiplied by 6, since they all have the same measure.
In this case the area of the cube is:
A=6×(15 cm)²
A= 1350 cm²
Then, deriving the area expression, you obtain that:
[tex]\frac{dA}{dx}[/tex]=6×2×side= 12×side
So:
dA=12×side×dx
Replacing the values in the previous expression you get:
dA= 12×15 cm×0.5cm
dA=90 cm²
The maximum error on the surface area is 90 cm².
The relative error is obtained by dividing the maximum error by the total volume:
[tex]\frac{dA}{A} =\frac{90 cm^{2}}{1350cm^{2} }[/tex]
[tex]\frac{dA}{A}[/tex]= 0.0667
The relative error on the surface area is 0.0667
Finally, the percentage error is calculated by multiplying the relative error by 100. In this case, the percentage error is calculated as:
0.0667× 100 = 10%
The percentage error on the volume is 6.67%.
Learn more:
- https://brainly.com/question/17016021?referrer=searchResults
- https://brainly.com/question/13370015?referrer=searchResults
- https://brainly.com/question/24202332?referrer=searchResults
- https://brainly.com/question/17134235?referrer=searchResults
