Can u help me in this pls

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

The expression 50w+400 represents the amount of money Jenny has in her savings account after an unknown number of weeks. How muc

nasty-shy [4]

Answer:

400 originally, and 50 dollars a week from there!!

Step-by-step explanation:

Since the 50 is followed by the variable "w", and the w stands for week, we can believe that Jenny makes 50 dollars a week.

Because the 400 was in the her account to begin with, we know this is the amount she originally had!

6 0

2 years ago

colby and jaquan are growing bacteria in an experiment in a laboratory. Colby starts with 50 bacteria in his culture and the num

kaheart [24]

To get started, we will use the general formula for bacteria growth/decay problems:

$A_{f} = A_{i} ( e^{kt} )$

where:
A_{f} = Final amount
A_{i} = Initial amount
k = growth rate constant
t = time

For doubling problems, the general formula can be shortened to:

$kt = ln(2)$

Now, we can use the shortened formula to calculate the growth rate constant of both bacteria:

Colby (1):
$k_{1} = ln(2)/t$
$k_{1} = ln(2)/2 = 0.34657$ per hour

Jaquan (2):
$k_{2} = ln(2)/t$
$k_{2} = ln(2)/3 = 0.23105$ per hour

Using Colby's rate constant, we can use the general formula to calculate for Colby's final amount after 1 day (24 hours).

Note: All units must be constant, so convert day to hours.

$A_{f1} = 50( e^{0.34657(24)})$
$A_{f1} = 204,800$

Remember that the final amount for both bacteria must be the same after 24 hours. Again, using the general formula, we can calculate the initial amount of bacteria that Jaquan needs:

$A_{f2} = 204,800 = A_{i2} ( e^{0.23105(24)} )$
$A_{i2} = 800$

3 0

2 years ago

Identify the characteristics of a severe storm

adelina 88 [10]

Answer:

According to the National Severe Storms Laboratory, a thunderstorm is a rain shower that is accompanied by thunder and lightning, and it may also involve string winds, sleet, or hail. A thunderstorm becomes severe when it develops hail larger than an inch in diameter, wind gusts in excess of 57.5 mph, or a tornado.

Step-by-step explanation:

8 0

2 years ago

Read 2 more answers

X is 4 less than a number y. What expression represents the value of x?

zhuklara [117]

Answer:

y - 4 = x

Step-by-step explanation:

4 less than y is y minus four, so therefore y - 4 = x

3 0

2 years ago

Read 2 more answers