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denpristay [2]
2 years ago
7

Nachelle is 1.45 meters tall. At 11 a.m., she measures the length of a tree's shadow to be 23.05 meters. She stands 18.9 meters

away from the tree, so that the tip of her shadow meets the tip of the tree's shadow. Find the height of the tree to the nearest hundredth of a meter.
Mathematics
1 answer:
GREYUIT [131]2 years ago
5 0

Step-by-step explanation:

this creates 2 similar triangles.

that means all angles are the same. and all the side lengths of one triangle correlate to the corresponding side lengths of the other triangle by the same multiplication factor.

her shadow is 23.05 - 18.9 = 4.15 m long.

the factor between the shadow lengths is then

4.15 × f = 23.05

f = 23.05 / 4.15 = 5.554216867...

the same factor now applies to the relation of the heights :

1.45 × 5.554216867... = 8.053614458... m

rounded the tree is 8.05 m tall.

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What length is the shorter leg of a right triangle that has hypotenuse 50 in long and its perimeter is 112 inches?
Brut [27]
The length of the shorter leg is 14 inches.

We will set up two equations for this; first, for the perimeter.  We will call the legs a and b:

a + b + 50 = 112

Subtract 50 from both sides:
a + b + 50 - 50 = 112 - 50
a + b = 62

We will isolate a, so we can use substitution.  Subtract b from both sides:
a + b - b = 62 - b
a = 62 - b

Now we will set up our Pythagorean theorem equation:
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a² + b² = 2500

We will substitute our value for a from the first equation into this one:
(62-b)² + b² = 2500
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Multiplying the binomials, we have:
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Combining like terms:
3844 - 124b + 2b² = 2500

To set it equal to 0 and solve, we subtract 2500 from both sides:
3844 - 124b + 2b² - 2500 = 2500 - 2500
1344 - 124b + 2b² = 0

In standard form, we have
2b² - 124b + 1344 = 0

Using the quadratic formula, 
\frac{--124\pm \sqrt{(-124)^2-4(2)(1344)}}{2(2)}
\\
\\=\frac{124\pm \sqrt{15376-10752}}{4}
\\
\\=\frac{124\pm \sqrt{4624}}{4}=\frac{124\pm 68}{4}=\frac{124+68}{4}\text{ or }\frac{124-68}{4}
\\
\\=\frac{192}{4}\text{ or }\frac{56}{4}=48\text{ or }14

The shorter leg is 14 and the longer one is 48.
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