What is the square root of 225?

The circumference of a sphere was measured to be 80 cm with a possible error of 0.5 cm. A) Use differentials to estimate the max

siniylev [52]

Answer:

A) The maximum error in the calculated surface area: $25cm^2$

Relative error: $0.013$

B) The maximum error in the calculated volume: $162cm^2$

Relative error: $0.019$

Step-by-step explanation:

A) The formula for the surface area is:

$A=4\pi r^2$

The measured value is the circumference which is equal to:

$C=2\pi r$

then the radius is:

$r=\frac{C}{2\pi}$

Substituting in the formula of the surface:

$A=4\pi(\frac{C}{2\pi})^2\\A=4\pi(\frac{C^2}{4\pi^2})\\A=\frac{C^2}{\pi}$

Using the formula to calculate the error:

$dy=f'(x)dx$

Where $x$ is the variable measured and $y$ is a function of $x$ ( $y=f(x)$ ).

$dA=f'(C)dC\\dA=\frac{2C^{(2-1)}}{\pi}dC\\dA=\frac{2C}{\pi}dC$

We have C=80cm and dC=0.5cm

$dA=\frac{2C}{\pi}dC\\dA=\frac{2(80)}{\pi}(0.5)\\dA=\frac{160}{\pi}(0.5)\\dA=50.9296(0.5)\\dA=25.4648\approx25cm^2$

The relative error is the maximum error divide by the total area. The total area is: $A=\frac{C^2}{\pi}=\frac{(80)^2}{\pi}=\frac{6400}{\pi}=2037.1833cm^2$

$\frac{dA}{A}=\frac{25.4648}{2037.1833} =0.0125\approx0.013$

B) The formula for the volume is:

$V=\frac{4}{3} \pi r^3$

Using $r=\frac{C}{2\pi}$

$V=\frac{4}{3} \pi r^3\\V=\frac{4}{3} \pi (\frac{C}{2\pi})^3\\V=\frac{4}{3} \pi (\frac{C^3}{8\pi^3})\\V=\frac{1}{3}(\frac{C^3}{2\pi^2})\\V=\frac{C^3}{6\pi^2}$

The maximum error is:

$dV=\frac{3C^{3-1}}{6\pi^2}dC\\dV=\frac{C^{2}}{2\pi^2}dC\\dV=\frac{(80)^{2}}{2\pi^2}(0.5)\\dV=\frac{6400}{2\pi^2}(0.5)\\dV=\frac{6400}{2\pi^2}(0.5)\\dV=(324.2278)(0.5)\\dV=162.1139\approx162cm^2$

The calculated volume is:

$V=\frac{C^3}{6\pi^2}\\V=\frac{(80)^3}{6\pi^2}\\V=\frac{512000}{6\pi^2}\\V=8646.0743$

The relative error is:

$\frac{dV}{V}=\frac{162.1139}{8646.0743}=0.0188\approx0.019$

3 0

2 years ago

a carpenter has a board that is 8 feet long he cuts of 2 pieces one piece is 3 and a half feet long the other is 2 and one feet

kakasveta [241]

3 1/2 I think
8-3 1/2=5 1/2
5 1/2-2=3 1/2

5 0

2 years ago

Read 2 more answers

I need help with this ^

Inga [223]

Answer:

23

Step-by-step explanation:

17 pluse 2 is 19

180 minus 19 is 161

add all the vs you get 7

so 161 divided by 7 is 23

6 0

2 years ago

Read 2 more answers

How many solutions does a triangle with values a=42, A=117 degrees, and b=34 have?

tatyana61 [14]

Answer:one solution

Step-by-step explanation:

Given

a=42

b=34

angle A=117

using sine rule

$\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}$

$\frac{42}{sin117}=\frac{34}{sinB}$

$sinB=\frac{34}{42}\times sin(117)$

sinB=0.7212

$B=46.161 ^{\circ}$

and A+B+C=180

$C=16.839^{\circ}$

Thus 2 triangles can be formed with given value

8 0

2 years ago

F(x)=(1/4)^x <br><br> what is the <br> y-intercept<br> common ratio

Vesna [10]

See attached graph.

The y-intercept is (0, 1), since the line is at that point when it intersects with the y-axis.

The common ratio is 1/4, since the y value multiplies by 1/4 for each unit on the x-axis.

6 0

3 years ago