To determine the total length of the two slides, we can use trigonometry, specifically the relationship between the angle, the opposite side, and the adjacent side of a right triangle.
Left Slide Calculation:
- Distance from the base = 64 cm
- Angle with the ground = 45°
Using the tangent function, we can find the height \(h\): \[ \tan(45°) = \frac{h}{64} \] Since \(\tan(45°) = 1\), we have: \[ 1 = \frac{h}{64} \implies h = 64 \text{ cm} \]
Now to find the length of the slide (hypotenuse \(l\)), we can use the Pythagorean theorem: \[ l^2 = 64^2 + 64^2 \] \[ l^2 = 4096 + 4096 = 8192 \] \[ l = \sqrt{8192} = 64\sqrt{2} \approx 90.5 \text{ cm} \]
Right Slide Calculation:
- Distance from the base = 106 cm
- Angle with the ground = 30°
Using the tangent function again to find the height \(h\): \[ \tan(30°) = \frac{h}{106} \] Since \(\tan(30°) = \frac{1}{\sqrt{3}} \approx 0.577\), we have: \[ \frac{1}{\sqrt{3}} = \frac{h}{106} \implies h = \frac{106}{\sqrt{3}} \approx 61.4 \text{ cm} \]
Now to find the length of the slide (hypotenuse \(l\)) using the Pythagorean theorem: \[ l^2 = 106^2 + h^2 \] Calculate \(h^2\): \[ h^2 = \left(\frac{106}{\sqrt{3}}\right)^2 = \frac{106^2}{3} \approx 3721.33 \] Calculate \(l^2\): \[ l^2 = 106^2 + \frac{106^2}{3} = 11236 + 3721.33 = 14957.33 \] \[ l = \sqrt{14957.33} \approx 122.3 \text{ cm} \]
Total Length of Both Slides: Now we can add both lengths together: \[ \text{Total Length} = 90.5 \text{ cm} + 122.3 \text{ cm} \approx 212.8 \text{ cm} \]
Rounding to the nearest tenth of a centimeter, the total length is approximately 212.9 cm.
Therefore, the answer is 212.9.