On a golf course, a flagstick casts a 150-inch shadow at the same time that a nearby tee casts a 5-inch shadow. If the tee is 2 3/4

Question

inches tall, how tall is the flagstick? Express your answer as a decimal if necessary.

Answer 1

We can find the height of the flagstick using the relationship between the heights of the objects and the lengths of their shadows. This relationship can be expressed using similar triangles.

Let \( h \) be the height of the flagstick. We can set up a proportion based on the heights and shadows:

\[ \frac{h}{150} = \frac{2.75}{5} \]

Now, we can cross-multiply to solve for \( h \):

\[ h \cdot 5 = 2.75 \cdot 150 \]

Calculating \( 2.75 \cdot 150 \):

\[ 2.75 \cdot 150 = 412.5 \]

Now we have:

\[ 5h = 412.5 \]

To find \( h \), divide both sides by 5:

\[ h = \frac{412.5}{5} = 82.5 \]

Thus, the height of the flagstick is

\[ \boxed{82.5} \text{ inches} \]

Answer 2

At the same time that an 11-inch-tall weathervane casts a 16-inch shadow, a nearby lightning rod casts a 28-inch shadow. How tall is the lightning rod? Express your answer as a decimal if necessary.

Answer 3

To find the height of the lightning rod, we can use the same method of proportions based on similar triangles.

Let \( h \) be the height of the lightning rod. We have the following relationship from the similar triangles:

\[ \frac{\text{height of weathervane}}{\text{shadow of weathervane}} = \frac{\text{height of lightning rod}}{\text{shadow of lightning rod}} \]

Plugging in the values we have:

\[ \frac{11}{16} = \frac{h}{28} \]

Now, we cross-multiply to solve for \( h \):

\[ 11 \cdot 28 = 16 \cdot h \]

Calculating \( 11 \cdot 28 \):

\[ 11 \cdot 28 = 308 \]

Now the equation is:

\[ 308 = 16h \]

To find \( h \), divide both sides by 16:

\[ h = \frac{308}{16} \]

Calculating this gives:

\[ h = 19.25 \]

Thus, the height of the lightning rod is

\[ \boxed{19.25} \text{ inches} \]

Answer 4

An obelisk casts a 27.5-meter shadow at the same time that a nearby monument casts a 5.5-meter shadow. If the monument is 4 meters tall, how tall is the obelisk? Express your answer as a decimal if necessary.

Answer 5

To find the height of the obelisk, we can use the proportion between the heights of the obelisk and the monument and the lengths of their shadows.

Let \( h \) be the height of the obelisk. The proportion can be set up as follows:

\[ \frac{\text{height of obelisk}}{\text{length of obelisk's shadow}} = \frac{\text{height of monument}}{\text{length of monument's shadow}} \]

Substituting the known values:

\[ \frac{h}{27.5} = \frac{4}{5.5} \]

Now, we cross-multiply to solve for \( h \):

\[ h \cdot 5.5 = 4 \cdot 27.5 \]

Calculating \( 4 \cdot 27.5 \):

\[ 4 \cdot 27.5 = 110 \]

So we have:

\[ 5.5h = 110 \]

Now, divide both sides by 5.5 to find \( h \):

\[ h = \frac{110}{5.5} \]

Calculating this gives:

\[ h = 20 \]

Thus, the height of the obelisk is

\[ \boxed{20} \text{ meters} \]

Answer 6

In the figure, the triangles are similar. What is the distance d across the pond? Express your answer as a decimal if necessary.



Pond and overlapping right triangles FGH and IJH. See long description.

Answer 7

To find the distance \(d\) across the pond using the property of similar triangles, we need to establish a proportion based on the dimensions of the triangles.

In similar triangles, the ratios of corresponding sides are equal. Therefore, if we have two triangles, triangle \(FGH\) and triangle \(IJH\), we can express the relationship as follows, assuming \(FGH\) is the larger triangle that includes the pond and \(IJH\) is the smaller one that shares a height \(H\):

\[ \frac{\text{Base of triangle } FGH}{\text{Height of triangle } FGH} = \frac{\text{Base of triangle } IJH}{\text{Height of triangle } IJH} \]

Let's use the variables:

• Let \(FG = d + x\) where \(d\) is the width across the pond and \(x\) is a known distance from point \(F\) to point \(I\).
• Let \(GH = H\) be the height of triangle \(FGH\).
• Let \(IJ = y\) be the base of triangle \(IJH\) (where \(y\) is a known measurement).

Then we could express it as:

\[ \frac{d + x}{H} = \frac{y}{h} \]

Where \(h\) is the height from points \(I\) or \(J\) to the base of triangle \(IJH\) (which can also be a known distance).

We can rearrange this equation to solve for \(d\):

\[ d + x = \frac{y}{h} \cdot H \]

\[ d = \frac{y}{h} \cdot H - x \]

Now you just need to plug in the known values of \(x\), \(y\), \(H\), and \(h\) to calculate \(d\).

Since you did not provide specific measurements or a diagram, please fill in the relevant values in the equation. If you provide those values, I can assist you further in calculating \(d\).