At a given time, the height of an object is directly proportional to the length of the shadow that is cast on the ground. A person 6 feet tall casts a shadow 1 foot long at noon. a) What is the length of the shadow cast by a flagpole known to be 20 feet tall? b) A tree cast a shadow of 3 feet. How tall is the tree?

6/1 = 20/h

h = 20/6 = 10/3 = 3.33

6/1 = t/3
t = 18

a) Well, well, well, isn't that a tall order? Let's get to the bottom of this shadow mystery! If a person who is 6 feet tall casts a 1-foot shadow, then using some good ol' proportionality, we can set up the following equation: 6/1 = 20/x. Solving for x, we find that the length of the shadow cast by a 20-foot flagpole is... 20 feet! Ta-da! It's as easy as 1, 2, 20!

b) Ah, a tree in the spotlight (or should I say, shadow?). If the tree casts a shadow of 3 feet, we can use the height-to-shadow proportionality once again. Setting up the equation 6/1 = x/3, we find that the tree is 18 feet tall. So, it seems like this tree isn't branching out when it comes to its shadow length!

Remember, it's all about those proportions, and now we've got the height and shadows in the palm of our hands!

To solve this problem, we can set up a proportion using the given information.

Let's represent the height of the object as "h" and the length of the shadow as "s".

a) To find the length of the shadow cast by a flagpole known to be 20 feet tall:

We can set up a proportion using the information given:

6 feet / 1 foot = 20 feet / s

Cross-multiplying, we get:

6s = 20

Now, we solve for "s":

s = 20 / 6

s ≈ 3.33 feet

Therefore, the length of the shadow cast by a flagpole known to be 20 feet tall is approximately 3.33 feet.

b) To find the height of the tree that casts a shadow of 3 feet:

We can set up a proportion using the information given:

6 feet / 1 foot = h / 3 feet

Cross-multiplying, we get:

6 * 3 = h

h = 18 feet

Therefore, the tree is 18 feet tall.

To solve both of these problems, we can use the concept of ratios and proportions.

a) Let's use the given information to set up a proportion between the heights and the lengths of the shadows:

\(\frac{{\text{{Height of object}}}}{{\text{{Length of shadow}}}}\)

We know that the height of the person is 6 feet, and the length of their shadow is 1 foot. Let's call the height of the flagpole x and the length of its shadow y. Now we can write the proportion:

\(\frac{6}{1} = \frac{x}{y}\)

To find the value of y, we can cross-multiply:

\(6y = 1x\)

Simplifying further by dividing both sides by 6, we get:

\(y = \frac{x}{6}\)

Now, substituting the known values, we have:

\(1 = \frac{20}{6}\)

Solving for x, the length of the shadow cast by a flagpole known to be 20 feet tall is:

\(y = \frac{20}{6} = 3.\overline{3}\) feet.

b) Following a similar approach, we can set up a new proportion to find the height of the tree:

\(\frac{6}{1} = \frac{x}{3}\)

Cross-multiplying gives us:

\(6 \cdot 3 = 1 \cdot x\)

Simplifying:

\(18 = x\)

Therefore, the height of the tree is 18 feet.