Questions: Aurinko paistaa pystysuoraa 15-metristä koivua vastaan 40 asteen kulmassa. Kuinka pitkä varjo piirtyy tasaiselle maanpinnalle? (Vinje: Piirrä kolmio.)

Solution Steps

To find the length of the shadow cast by the tree, we can use trigonometry. Specifically, we can use the tangent function, which relates the angle of elevation to the opposite side (height of the tree) and the adjacent side (length of the shadow). The formula is:

\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

Rearranging to solve for the length of the shadow (adjacent side):

\[ \text{adjacent} = \frac{\text{opposite}}{\tan(\theta)} \]

Given:

• Height of the tree (opposite side) = 15 meters
• Angle of elevation (\(\theta\)) = 40 degrees

We can now calculate the length of the shadow.

We are given the height of the tree as \( h = 15 \) meters and the angle of elevation as \( \theta = 40^\circ \).

To use trigonometric functions, we convert the angle from degrees to radians: \[ \theta_{\text{radians}} = \frac{40 \times \pi}{180} \approx 0.6981 \]

We apply the tangent function to find the length of the shadow \( s \): \[ \tan(\theta) = \frac{h}{s} \] Rearranging gives: \[ s = \frac{h}{\tan(\theta)} \]

Substituting the known values: \[ s = \frac{15}{\tan(0.6981)} \approx 17.8763 \] Rounding to four significant digits, we find: \[ s \approx 17.88 \text{ meters} \]

Final Answer