A tree farmer used the formula $$ V=0.5 d^{2} h $$ to estimate the volume, $$ V $$ cubie metres, of a tree with height and its volume is $$ 230 \mathrm{~m}^{3} $$. What is the mean trunk diameter of this tree to the nearest metre? ander,

Question

UpStudy AI · Accepted Answer

We are given the formula

$$
V=0.5\,d^2\,h
$$

and for a tree of volume $$ V=230\text{ m}^3 $$ and height $$ h $$ (in metres) the formula becomes

$$
230=0.5\,d^2\,h.
$$

Suppose the tree’s height is known; for example, if the tree’s height is $$ h=20 $$ metres, then substituting in we have

$$
230=0.5\,d^2\,(20).
$$

Step 1. Simplify the right-hand side:

$$
0.5 \times 20=10,
$$

so the equation becomes

$$
230=10\,d^2.
$$

Step 2. Solve for $$ d^2 $$ by dividing both sides by 10:

$$
d^2=\frac{230}{10}=23.
$$

Step 3. Take the square root of both sides to find $$ d $$:

$$
d=\sqrt{23}\approx4.80.
$$

Step 4. Rounding to the nearest metre, the mean trunk diameter is

$$
d\approx5\text{ m.}
$$

Thus, with a height of $$20$$ metres, the tree’s mean trunk diameter is $$5$$ metres.

If the tree’s height is different from $$20$$ metres, then the general solution for $$ d $$ is

$$
d=\sqrt{\frac{2\times230}{h}}=\sqrt{\frac{460}{h}},
$$

and you would substitute the given height value into this formula to determine $$ d $$ and round to the nearest metre.
To find the mean trunk diameter of the tree, use the formula $$ V = 0.5 \times d^2 \times h $$. Given $$ V = 230 $$ m³ and assuming a height $$ h = 20 $$ m, plug the values into the formula: $$ 230 = 0.5 \times d^2 \times 20 $$ Simplify: $$ 230 = 10 \times d^2 $$ Solve for $$ d^2 $$: $$ d^2 = \frac{230}{10} = 23 $$ Take the square root: $$ d \approx \sqrt{23} \approx 4.80 \text{ m} $$ Rounded to the nearest metre, the mean trunk diameter is approximately **5 metres**.

A tree farmer used the formula to estimate the volume, cubie metres, of a tree with height
and its volume is . What is the mean trunk diameter of this tree to the nearest metre?
ander,

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Algebra Questions

A tree farmer used the formula to estimate the volume, cubie metres, of a tree with height and its volume is . What is the mean trunk diameter of this tree to the nearest metre? ander,