A ladder 20 feet long leans against a building forming and angle of 46 degrees with the level ground. To the nearest foot how fa
1 answer:
Answer:
The distance of the foot of the ladder to the building is 14 ft.
Step-by-step explanation:
The length of ladder = 20 ft
Angle formed by ladder with level ground, θ = 46
We are required to find out the distance of the foot of the ladder from the building
The above question can be found out by using trigonometric relations as follows;
The adjacent side of the right triangle formed by the ladder the building and the ground is the distance of the foot of the ladder from the building
The hypotenuse side is the length of the ladder = 20 ft
Therefore;
Adjacent side of triangle = Hypotenuse × cosθ
∴ Distance of the foot of the ladder from the building = Hypotenuse × cosθ
Distance of the foot of the ladder from the building = 20 ft × cos(56)
Distance of the foot of the ladder from the building = 13.893 ft
To the nearest foot, the distance of the foot of the ladder to the building = 14 ft.
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<span>The <u>correct answer</u> is:
x=64/15, y=136/15.
Explanation:
We will use the Pythagorean theorem to solve this.
For the smaller triangle, the legs of the triangle are x and 8, and the hypotenuse is 8. In the Pythagorean theorem, this gives us:
x</span>²<span>+8</span>²<span>=y</span>²
<span>
Simplifying, we have:
x</span>²<span>+64=y</span>²<span>.
For the large triangle, the legs of the triangle are y and 17, and the hypotenuse is 15+x. In the Pythagorean theorem, this gives us:
y</span>²<span>+17</span>²<span>=(15+x)</span>²<span>.
Simplifying the left hand side we have:
y</span>²<span>+289 = (15+x)</span>²<span>.
In order to simplify the right hand side, we write (15+x)</span>²<span> as the product of two binomials:
y</span>²<span>+289=(15+x)(15+x).
Multiplying the right hand side, we have:
y</span>²<span>+289=15*15+15*x+x*15+x*x
y</span>²<span>+289=225+15x+15x+x</span>²
<span>y</span>²<span>+289=225+30x+x</span>²<span>.
We can now substitute the value from the smaller triangle in place of y</span>²<span>:
x</span>²<span>+64+289=225+30x+x</span>²<span>.
Combining like terms<span>, we have:
x</span></span>²<span><span>+353=225+30x+x</span></span>²<span><span>.
Subtracting x</span></span>²<span><span> from both sides, we have:
x</span></span>²<span><span>+353-x</span></span>²<span><span>=225+30x+x</span></span>²<span><span>-x</span></span>²
<span><span>353=225+30x.
Subtract 225 from both sides:
353-225=225+30x-225
128=30x.
Divide both sides by 30:
128/30=30x/30
128/30 = x.
Both the numerator and denominator of this fraction are divisible by 2:
(128/2)/(30/2)=64/15=x.
Now we use this to find y; in our Pythagorean theorem equation from earlier, we have:
x</span></span>²<span><span>+64=y</span></span>²<span><span>.
Using our value for x, we have:
(64/15)</span></span>²<span><span>+64=y</span></span>²<span><span>.
Squaring the fraction, we have:
4096/225+64=y</span></span>²<span><span>.
In order to write 64 as a fraction using 225 as the denominator, we multiply:
(64*225)/225=14400/225.
This gives us:
4096/225+14400/225=y</span></span>²
<span><span>
Combining like terms,
18496/225=y</span></span>²<span><span>.
Taking the square root of both sides, we have:
</span></span>
<span><span>
</span></span>
x=64/15, y=136/15.
Explanation:
We will use the Pythagorean theorem to solve this.
For the smaller triangle, the legs of the triangle are x and 8, and the hypotenuse is 8. In the Pythagorean theorem, this gives us:
x</span>²<span>+8</span>²<span>=y</span>²
<span>
Simplifying, we have:
x</span>²<span>+64=y</span>²<span>.
For the large triangle, the legs of the triangle are y and 17, and the hypotenuse is 15+x. In the Pythagorean theorem, this gives us:
y</span>²<span>+17</span>²<span>=(15+x)</span>²<span>.
Simplifying the left hand side we have:
y</span>²<span>+289 = (15+x)</span>²<span>.
In order to simplify the right hand side, we write (15+x)</span>²<span> as the product of two binomials:
y</span>²<span>+289=(15+x)(15+x).
Multiplying the right hand side, we have:
y</span>²<span>+289=15*15+15*x+x*15+x*x
y</span>²<span>+289=225+15x+15x+x</span>²
<span>y</span>²<span>+289=225+30x+x</span>²<span>.
We can now substitute the value from the smaller triangle in place of y</span>²<span>:
x</span>²<span>+64+289=225+30x+x</span>²<span>.
Combining like terms<span>, we have:
x</span></span>²<span><span>+353=225+30x+x</span></span>²<span><span>.
Subtracting x</span></span>²<span><span> from both sides, we have:
x</span></span>²<span><span>+353-x</span></span>²<span><span>=225+30x+x</span></span>²<span><span>-x</span></span>²
<span><span>353=225+30x.
Subtract 225 from both sides:
353-225=225+30x-225
128=30x.
Divide both sides by 30:
128/30=30x/30
128/30 = x.
Both the numerator and denominator of this fraction are divisible by 2:
(128/2)/(30/2)=64/15=x.
Now we use this to find y; in our Pythagorean theorem equation from earlier, we have:
x</span></span>²<span><span>+64=y</span></span>²<span><span>.
Using our value for x, we have:
(64/15)</span></span>²<span><span>+64=y</span></span>²<span><span>.
Squaring the fraction, we have:
4096/225+64=y</span></span>²<span><span>.
In order to write 64 as a fraction using 225 as the denominator, we multiply:
(64*225)/225=14400/225.
This gives us:
4096/225+14400/225=y</span></span>²
<span><span>
Combining like terms,
18496/225=y</span></span>²<span><span>.
Taking the square root of both sides, we have:
</span></span>
</span></span>