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3 years ago

A sphere has a radius of 11 feet. A second sphere has a radius of 8 feet. what is the difference of the volume of thespheres

Mathematics

1 answer:

Veronika [31]3 years ago

7 0

The difference between the volume of the spheres is 3428.88 cubic feet

Explanation:

Given that one sphere has a radius of 11 feet.

A second sphere has a radius of 8 feet.

Volume of the 1st sphere:

The formula to determine the volume of the sphere is given by

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

Volume of the 1st sphere is given by

$V=\frac{4}{3}(3.14)(11)^3$

$V=\frac{4}{3}(3.14)(1331)$

$V=\frac{16717.36}{3}$

$V=5572.45$

The volume of the 1st sphere is 5572.45 cubic feet.

Volume of the 2nd sphere:

Volume of the 2nd sphere is given by

$V=\frac{4}{3}(3.14)(8)^3$

$V=\frac{4}{3}(3.14)(512)$

$V=\frac{6430.72}{3}$

$V=2143.57$

The volume of the 2nd sphere is 2143.57 cubic feet.

Difference between the volume of the two spheres:

Difference = Volume of the 1st sphere - Volume of the 2nd sphere

= 5572.45 - 2143.57

Difference = 3428.88 cubic feet.

Hence, the difference between the volume of the spheres is 3428.88 cubic feet.

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$z=\frac{1.02-1}{\frac{0.04}{\sqrt{40}}}=3.162$

$p_v =2*P(z>3.162)=0.0016$

If we compare the p value and the significance level given $\alpha=0.05$ we see that $p_v$ so we can conclude that we have enough evidence to to reject the null hypothesis, so we can conclude that the true mean is different from 1 pound at 5% of signficance.

Step-by-step explanation:

Data given and notation

$\bar X=1.02$ represent the sample mean

$\sigma=0.04$ represent the population standard deviation

$n=40$ sample size

$\mu_o =1$ represent the value that we want to test

$\alpha=0.05$ represent the significance level for the hypothesis test.

z would represent the statistic (variable of interest)

$p_v$ represent the p value for the test (variable of interest)

State the null and alternative hypotheses.

We need to conduct a hypothesis in order to check if the true mean is equal to 1 pound or no :

Null hypothesis: $\mu =1$

Alternative hypothesis: $\mu \neq 1$

Since we know the population deviation, is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:

$z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}}$ (1)

z-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".

Calculate the statistic

We can replace in formula (1) the info given like this:

$z=\frac{1.02-1}{\frac{0.04}{\sqrt{40}}}=3.162$

P-value

Since is a two-sided test the p value would be:

$p_v =2*P(z>3.162)=0.0016$

Conclusion

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