Math  /  Geometry

QuestionCarolina is hanging Christmas lights from a tree in the center of her yard. She wants the lights to be straight similar to the diagram below. She knows the lights are 30 feet ( ft .) long. She stands 5512ft5 \frac{5}{12} \mathrm{ft}. tall and holds the lights 616ft6 \frac{1}{6} \mathrm{ft}. away from the point where they will secure into the ground. Using the diagram, how far up the tree should she place the hook to hold the lights? \square ft.

Studdy Solution

STEP 1

1. The lights form a straight line from the hook on the tree to the ground.
2. Carolina's height and the distance she holds the lights from the tree form a right triangle with the tree.
3. We need to find the height on the tree where the hook should be placed.

STEP 2

1. Convert mixed numbers to improper fractions.
2. Use the Pythagorean theorem to find the height where the hook should be placed.
3. Solve for the height on the tree.

STEP 3

Convert Carolina's height and the distance she holds the lights from mixed numbers to improper fractions.
Carolina's height: 5512 5 \frac{5}{12} ft. =5×12+512=6512 ft. = \frac{5 \times 12 + 5}{12} = \frac{65}{12} \text{ ft.}
Distance from the tree: 616 6 \frac{1}{6} ft. =6×6+16=376 ft. = \frac{6 \times 6 + 1}{6} = \frac{37}{6} \text{ ft.}

STEP 4

Use the Pythagorean theorem to find the height on the tree where the hook should be placed. The Pythagorean theorem states a2+b2=c2 a^2 + b^2 = c^2 , where c c is the hypotenuse (length of the lights), a a is the height from the ground to the hook, and b b is the distance from the tree.
Given: c=30 ft. c = 30 \text{ ft.} b=376 ft. b = \frac{37}{6} \text{ ft.}
We need to find a a .

STEP 5

Rearrange the Pythagorean theorem to solve for a a :
a2=c2b2 a^2 = c^2 - b^2 a2=302(376)2 a^2 = 30^2 - \left(\frac{37}{6}\right)^2

STEP 6

Calculate c2 c^2 and b2 b^2 :
c2=302=900 c^2 = 30^2 = 900 b2=(376)2=136936 b^2 = \left(\frac{37}{6}\right)^2 = \frac{1369}{36}

STEP 7

Substitute the values into the equation for a2 a^2 :
a2=900136936 a^2 = 900 - \frac{1369}{36}
Convert 900 to a fraction with a denominator of 36:
900=3240036 900 = \frac{32400}{36}
a2=3240036136936 a^2 = \frac{32400}{36} - \frac{1369}{36} a2=32400136936 a^2 = \frac{32400 - 1369}{36} a2=3103136 a^2 = \frac{31031}{36}

STEP 8

Take the square root of both sides to solve for a a :
a=3103136 a = \sqrt{\frac{31031}{36}}

STEP 9

Calculate the square root:
a310316 a \approx \frac{\sqrt{31031}}{6}
Using a calculator:
a176.1629.35 ft. a \approx \frac{176.1}{6} \approx 29.35 \text{ ft.}
The height on the tree where the hook should be placed is approximately:
29.35 ft. \boxed{29.35} \text{ ft.}

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