Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Join our Q&A platform to connect with experts dedicated to providing precise answers to your questions in different areas. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
Answer:
The volume is 40.96 cubic feet
Step-by-step explanation:
Given
[tex]V\ \alpha\ h * d^2[/tex]
Where
[tex]V = Volume\\ h = height\\ d = distance[/tex]
[tex]V = 15.84ft^3; h =22ft; d = 3ft[/tex]
Required
The volume when h = 32 and d = 4
[tex]V\ \alpha\ h * d^2[/tex]
Express as an equation
[tex]V = khd^2[/tex]
Where k is the constant of variation.
Make k the subject
[tex]k = \frac{V}{hd^2}[/tex]
Substitute [tex]V = 15.84ft^3; h =22ft; d = 3ft[/tex]
[tex]k = \frac{15.84}{22 * 3^2}[/tex]
[tex]k = \frac{15.84}{22 * 9}[/tex]
[tex]k = \frac{15.84}{198}[/tex]
[tex]k = 0.08[/tex]
To solve for V when h = 32 and d = 4, we have:
[tex]V = khd^2[/tex]
[tex]V = 0.08 * 32 * 4^2[/tex]
[tex]V = 0.08 * 32 * 16[/tex]
[tex]V = 40.96[/tex]
The volume is 40.96 cubic feet
The volume of the wood obtained from a tree that is 32 feet tall having a measurement of 4 feet around the trunk is [tex]40.96 \: \rm ft^3[/tex]
What is directly proportional and inversely proportional relationship?
Let there are two variables p and q
Then, p and q are said to be directly proportional to each other if
[tex]p = kq[/tex]
where k is some constant number called constant of proportionality.
This directly proportional relationship between p and q is written as
[tex]p \propto q[/tex] where that middle sign is the sign of proportionality.
In a directly proportional relationship, increasing one variable will increase another.
Now let m and n are two variables.
Then m and n are said to be inversely proportional to each other if
[tex]m = \dfrac{c}{n} \\\\ \text{or} \\\\ n = \dfrac{c}{m}[/tex]
(both are equal)
where c is a constant number called constant of proportionality.
This inversely proportional relationship is denoted by
[tex]m \propto \dfrac{1}{n} \\\\ \text{or} \\\\n \propto \dfrac{1}{m}[/tex]
As visible, increasing one variable will decrease the other variable if both are inversely proportional.
For this case, we have:
V = volume of the wood varying jointly as height of the tree and square of the distance around the trunk.
Let we take:
- r = distance of the wood around the trunk (in feet)
- h = height of the tree ( in feet)
V will of course increase as r or h increases, and as it varies jointly, thus:
[tex]V \propto h r^2[/tex]
Let the constant of proportionality be 'k', then:
[tex]V = kr^2h[/tex] (in cubic foot)
For case 1, it was specified that V = 15.84 cubic foot when r = 3 ft and h = 22 ft,
Putting these values gives us:
[tex]15.84 = k(3)^2(22)\\\\k = \dfrac{15.84}{198} = 0.08[/tex]
Thus, we have:
[tex]\rm V = 0.08r^2 h \: \rm ft^3[/tex]
Assuming second wood was also obtained from same kind of tree, we get the volume of the wood for h = 32 and r = 4 feet as:
[tex]V = 0.08 (4)^2 (32) = 40.96 \: \rm ft^3[/tex]
Thus, the volume of the wood obtained from a tree that is 32 feet tall having a measurement of 4 feet around the trunk is [tex]40.96 \: \rm ft^3[/tex]
Learn more about proportionality here:
https://brainly.com/question/13082482
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
