$33$ David has just finished building his tree house and still needs to buy a ladder to be attached to the ledge of the treehouse and anchored at a point on the ground, as modeled below. David is standing $1$ . $3$ meters from the stilt supporting the treehouse. This is the point on the ground where he has decided to anchor the ladder. The angle of elevation from his eye level to the bottom of the treehouse is $56$ degrees. David's eye level is $1$ . $5$ meters above the ground. Determine and state the minimum length ladder, to the nearest tenth of a meter, that David will need to buy for his treehouse.

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Pythagorean Theorem and its converse

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33

David has just finished building his tree house and still needs to buy a ladder to be attached to the ledge of the treehouse and anchored at a point on the ground, as modeled below. David is standing

1

3

meters from the stilt supporting the treehouse. This is the point on the ground where he has decided to anchor the ladder. The angle of elevation from his eye level to the bottom of the treehouse is

56

degrees. David's eye level is

1

5

meters above the ground. Determine and state the minimum length ladder, to the nearest tenth of a meter, that David will need to buy for his treehouse.

David's Eye Level and Angle: David's eye level is $1.5$ meters above the ground, and the angle of elevation is $56$ degrees. We need to find the length of the ladder, which is the hypotenuse of the right triangle formed by David's eye level, the distance from the stilt, and the ladder itself.
Trigonometry Calculation: Using trigonometry, specifically the tangent function, which relates the opposite side to the adjacent side in a right triangle: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ . Here, $\theta$ is $56$ degrees, the opposite side is the height of the treehouse from David's eye level, and the adjacent side is the distance from the stilt to where the ladder will be anchored, which is $1.3$ meters.
Calculate Height of Treehouse: First, calculate the height of the treehouse from David's eye level using the tangent function: $\tan(56^\circ) = \frac{\text{height}}{1.3}$ .
Total Height Calculation: $\text{height} = \tan(56) \times 1.3$ .
Pythagorean Theorem Application: $\text{height} = 2.1445 \times 1.3$ .
Calculate Ladder Length: height = $2.788$ meters.
Calculate Ladder Length: height = $2.788$ meters.Now, we need to add David's eye level to the height we just calculated to get the total height from the ground to the bottom of the treehouse: total height = height + David's eye level.
Calculate Ladder Length: height = $2.788$ meters.Now, we need to add David's eye level to the height we just calculated to get the total height from the ground to the bottom of the treehouse: total height = height + David's eye level.total height = $2.788 + 1.5$ .
Calculate Ladder Length: height = $2.788$ meters. Now, we need to add David's eye level to the height we just calculated to get the total height from the ground to the bottom of the treehouse: total height = height + David's eye level. total height = $2.788 + 1.5$ . total height = $4.288$ meters.
Calculate Ladder Length: height = $2.788$ meters. Now, we need to add David's eye level to the height we just calculated to get the total height from the ground to the bottom of the treehouse: total height = height + David's eye level. total height = $2.788 + 1.5$ . total height = $4.288$ meters. Now we use the Pythagorean theorem to find the hypotenuse (the ladder length). The total height is one leg, and the distance from the stilt ( $1.3$ meters) is the other leg.
Calculate Ladder Length: height = $2.788$ meters. Now, we need to add David's eye level to the height we just calculated to get the total height from the ground to the bottom of the treehouse: total height = height + David's eye level. total height = $2.788 + 1.5$ . total height = $4.288$ meters. Now we use the Pythagorean theorem to find the hypotenuse (the ladder length). The total height is one leg, and the distance from the stilt ( $1.3$ meters) is the other leg. $\text{ladder}^2 = \text{total height}^2 + \text{distance from stilt}^2$ .
Calculate Ladder Length: height = $2.788$ meters. Now, we need to add David's eye level to the height we just calculated to get the total height from the ground to the bottom of the treehouse: total height = height + David's eye level. total height = $2.788 + 1.5$ . total height = $4.288$ meters. Now we use the Pythagorean theorem to find the hypotenuse (the ladder length). The total height is one leg, and the distance from the stilt ( $1.3$ meters) is the other leg. $\text{ladder}^2 = \text{total height}^2 + \text{distance from stilt}^2$ . $\text{ladder}^2 = 4.288^2 + 1.3^2$ .
Calculate Ladder Length: height = $2.788$ meters. Now, we need to add David's eye level to the height we just calculated to get the total height from the ground to the bottom of the treehouse: total height = height + David's eye level. total height = $2.788 + 1.5$ . total height = $4.288$ meters. Now we use the Pythagorean theorem to find the hypotenuse (the ladder length). The total height is one leg, and the distance from the stilt ( $1.3$ meters) is the other leg. $\text{ladder}^2 = \text{total height}^2 + \text{distance from stilt}^2$ . $\text{ladder}^2 = 4.288^2 + 1.3^2$ . $\text{ladder}^2 = 18.386 + 1.69$ .
Calculate Ladder Length: height = $2.788$ meters. Now, we need to add David's eye level to the height we just calculated to get the total height from the ground to the bottom of the treehouse: total height = height + David's eye level. total height = $2.788 + 1.5$ . total height = $4.288$ meters. Now we use the Pythagorean theorem to find the hypotenuse (the ladder length). The total height is one leg, and the distance from the stilt ( $1.3$ meters) is the other leg. $\text{ladder}^2 = \text{total height}^2 + \text{distance from stilt}^2$ . $\text{ladder}^2 = 4.288^2 + 1.3^2$ . $\text{ladder}^2 = 18.386 + 1.69$ . $\text{ladder}^2 = 20.076$ .
Calculate Ladder Length: height = $2.788$ meters. Now, we need to add David's eye level to the height we just calculated to get the total height from the ground to the bottom of the treehouse: total height = height + David's eye level. total height = $2.788 + 1.5$ . total height = $4.288$ meters. Now we use the Pythagorean theorem to find the hypotenuse (the ladder length). The total height is one leg, and the distance from the stilt ( $1.3$ meters) is the other leg. $\text{ladder}^2 = \text{total height}^2 + \text{distance from stilt}^2$ . $\text{ladder}^2 = 4.288^2 + 1.3^2$ . $\text{ladder}^2 = 18.386 + 1.69$ . $\text{ladder}^2 = 20.076$ . $\text{ladder} = \sqrt{20.076}$ .
Calculate Ladder Length: height = $2.788$ meters. Now, we need to add David's eye level to the height we just calculated to get the total height from the ground to the bottom of the treehouse: total height = height + David's eye level. total height = $2.788 + 1.5$ . total height = $4.288$ meters. Now we use the Pythagorean theorem to find the hypotenuse (the ladder length). The total height is one leg, and the distance from the stilt ( $1.3$ meters) is the other leg. $\text{ladder}^2 = \text{total height}^2 + \text{distance from stilt}^2$ . $\text{ladder}^2 = 4.288^2 + 1.3^2$ . $\text{ladder}^2 = 18.386 + 1.69$ . $\text{ladder}^2 = 20.076$ . $\text{ladder} = \sqrt{20.076}$ . $\text{ladder} = 4.48$ meters.
Calculate Ladder Length: height = $2.788$ meters. Now, we need to add David's eye level to the height we just calculated to get the total height from the ground to the bottom of the treehouse: total height = height + David's eye level. total height = $2.788 + 1.5$ . total height = $4.288$ meters. Now we use the Pythagorean theorem to find the hypotenuse (the ladder length). The total height is one leg, and the distance from the stilt ( $1.3$ meters) is the other leg. $\text{ladder}^2 = \text{total height}^2 + \text{distance from stilt}^2$ . $\text{ladder}^2 = 4.288^2 + 1.3^2$ . $\text{ladder}^2 = 18.386 + 1.69$ . $\text{ladder}^2 = 20.076$ . $\text{ladder} = \sqrt{20.076}$ . $\text{ladder} = 4.48$ meters. Round the ladder length to the nearest tenth of a meter: $2.788 + 1.5$ $0$ meters.