Which equation shows the correct use of the Distributive Property?
2 answers:
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Answer:
The radius of the base of the cone is 2 units
The slant height of the cone is 4 units
The height of the cone is 2√3 units
The volume of the cone is units³
Step-by-step explanation:
* Lets revise the total surface area and the lateral area of a cone
- The lateral area of cone = π r l , where r is the radius of the base
and l is the slant height of the cone
- The surface area of the cone = π r l + π r², where π r l is the lateral
area and π r² is the base area
- The cone has three dimensions radius (r) , height (h) , slant height (l)
- r , h , l formed right triangle, r , h are its legs and l is its hypotenuse,
then l² = r² + h²
- The volume of the con = (π r² h)
* Now lets solve the problem
- We will use the total area to find the radius of the base
∵ TA = 12π
∵ TA = LA + πr²
∵ LA = 8π
- Substitute the value of the lateral area in the total area
∴ 12π = 8π + π r² ⇒ subtract 8π from both sides
∴ 12π - 8π = π r²
∴ 4π = π r² ⇒ divide both sides by π
∴ r² = 4 ⇒ take square root for both sides
∴ r = 2
* The radius of the base of the cone is 2 units
- We will use the lateral area to find the slant height
∵ LA = π r l
∵ LA = 8π
∵ r = 2
∴ π (2) l = 8π ⇒ divide both sides by π
∴ 2 l = 8 ⇒ divide both sides by 2
∴ l = 4
* The slant height of the cone is 4 units
- Use the rule l² = r² + h² to find the height of the cone
∵ r = 2 and l = 4
∵ l² = r² + h²
∴ (4)² = (2)² + h²
∴ 16 = 4 + h² ⇒ subtract 4 from both sides
∴ 12 = h² ⇒ take square root for both sides
∴ h = √12 = 2√3
* The height of the cone is 2√3 units
∵ The volume of the con = (π r² h)
∵ r = 2 and h = 2√3
∴ V = (π × 2² × 2√3) =
(π × 4 × 2√3) =
(π × 8√3)
∴ V =
* The volume of the cone is units³
Answer:
(a) <em>H₀</em>: <em>μ</em> = 5 vs. <em>Hₐ</em>: <em>μ</em> ≠ 5.
(b) The test statistic value is -3.67.
(c) The <em>p</em>-value of the test is 0.0025.
(d) The mean amount of time spent in the shower by an adult is different from 5 minutes.
Step-by-step explanation:
In this case we need to test whether the mean amount of time that college students spend in the shower is significantly different from 5 minutes.
The information provided is:
(a)
The hypothesis for the test can be defined as follows:
<em>H₀</em>: The mean amount of time spent in the shower by an adult is 5 minutes, i.e. <em>μ</em> = 5.
<em>Hₐ</em>: The mean amount of time spent in the shower by an adult is different from 5 minutes, i.e. <em>μ</em> ≠ 5.
(b)
As the population standard deviation is not known we will use a <em>t</em>-test for single mean.
Compute the test statistic value as follows:
Thus, the test statistic value is -3.67.
(c)
Compute the <em>p</em>-value of the test as follows:
*Use a <em>t</em>-table.
Thus, the <em>p</em>-value of the test is 0.0025.
(d)
Decision rule:
If the <em>p</em>-value of the test is less than the significance level then the null hypothesis will be rejected and vice-versa.
<em>p</em>-value = 0.0025 < <em>α</em> = 0.01
The null hypothesis will be rejected at 1% level of significance.
Thus, concluding that the mean amount of time spent in the shower by an adult is different from 5 minutes.