Question

a playground has two slides on either side. the side on the left is situated at a 45 - degree angle to the ground and is 68 centimeters away from the base of the playground. the slide on the right is positioned at a 30 - degree angle to the ground and is 112 centimeters away from the base. what is the total length of both slides on this playground, rounded to the nearest tenth of a centimeter?
options: 96.2, 123.3, 197.3, 225.5

Explanation:

Step1: Calculate length of left - side slide

For the left - side slide which makes a 45 - degree angle with the ground and is 68 cm away from the base. Using the sine function in a right - triangle $\sin\theta=\frac{opposite}{hypotenuse}$. In a 45 - 45 - 90 triangle, if the adjacent side to the 45 - degree angle is $x = 68$ cm, and $\sin45^{\circ}=\frac{\sqrt{2}}{2}$, and since $\sin45^{\circ}=\frac{h}{l}$ (where $h$ is the height and $l$ is the length of the slide), and also $\cos45^{\circ}=\frac{x}{l}$, and $\cos45^{\circ}=\sin45^{\circ}=\frac{\sqrt{2}}{2}$, the length of the left - side slide $l_1=\frac{68}{\cos45^{\circ}}=68\sqrt{2}\approx68\times1.414 = 96.152$ cm.

Step2: Calculate length of right - side slide

For the right - side slide which makes a 30 - degree angle with the ground and is 112 cm away from the base. Using the cosine function in a right - triangle $\cos\theta=\frac{adjacent}{hypotenuse}$. Let the length of the right - side slide be $l_2$. We know that $\cos30^{\circ}=\frac{\sqrt{3}}{2}$ and $\cos30^{\circ}=\frac{112}{l_2}$, so $l_2=\frac{112}{\cos30^{\circ}}=\frac{112}{\frac{\sqrt{3}}{2}}=\frac{224}{\sqrt{3}}\approx\frac{224}{1.732}\approx129.33$ cm.

Step3: Calculate total length

The total length of both slides $L = l_1 + l_2\approx96.152+129.33 = 225.482\approx225.5$ cm.

Answer:

225.5