The volume of wood in a tree varies jointly as the height of the tree and the
square of the distance around the tree trunk. If the volume of wood is 15.84 cubic
feet when the height is 22 feet and the distance around the trunk is 3 feet, what is
the volume of wood obtained from a tree that is 32 feet tall having a measurement
of 4 feet around the trunk?



please help :))

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]

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:

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:

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