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11111nata11111 [884]
3 years ago
12

A 9.85 m ladder is placed against a wall. The height to the top of the ladder is 5 m more than the distance between the wall and

the foot of the ladder. Find the height to the top and the distance between the wall and the foot of the ladder.​ find base and height
base...... m
height.....m
Mathematics
1 answer:
Dafna1 [17]3 years ago
5 0

Given:

Length of the ladder = 9.85 m

The height to the top of the ladder is 5 m more than the distance between the wall and the foot of the ladder.

To find:

The height to the top and the distance between the wall and the foot of the ladder.​

Solution:

let x be the distance between the wall and the foot of the ladder. Then the height to the top of the ladder is (x+5).

Pythagoras theorem: In a right angle triangle,

Hypotenuse^2=Base^2+Perpendicular^2

In the given situation, hypotenuse is the length of ladder, i.e., 9.85 m. The base is x m and the height is (x+5) m.

Using the Pythagoras theorem, we get

(9.85)^2=x^2+(x+5)^2

97.0225=x^2+x^2+10x+25

0=2x^2+10x+25-97.0225

0=2x^2+10x-72.0225

Here, a=2, b=10,c=-72.0225. Using the quadratic formula, we get

x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}

x=\dfrac{-10\pm \sqrt{(10)^2-4(2)(-72.0225)}}{2(2)}

x=\dfrac{-10\pm \sqrt{676.18}}{4}

Approximating the value, we get

x=\dfrac{-10\pm 26}{4}

x=\dfrac{-10+26}{4},\dfrac{-10-26}{4}

x=\dfrac{16}{4},\dfrac{-36}{4}

x=4,-9

Distance cannot be negative so x\neq -9.

Now we have x=4

x+5=4+5

x+5=9

Therefore, the base is 4 m and the height is 9 m.

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1) 2 nonreal complex roots

2) 1 Real Solution

3) 16

4) Reflected, narrower by a factor of 2/5, slides right 4 units and slides up 6 (units)

Step-by-step explanation:

1) The graph does not intercept the x-axis, therefore, there are no real solutions at the point y = 0

We get;

y = a·x² + b·x + c

At y = 6, x = -2

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6 = a·(-2)² - 2·b + c = 4·a - 2·b + c

6 = 4·a - 2·b + c...(1)

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Similarly, we have;

At y = 8, x = -4

8 = a·(-4)² - 4·b + c = 16·a - 4·b + 8

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From equation (1), (2) and (3), we have;

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The equation is therefor;

y = (1/2)·x² + 2·x + 8

Solving we get;

x = (-2 ± √(2² - 4 × (1/2) × 8))/(2 × (1/2))

x =( -2 ± √(-12))/1 = -2 ± √(-12)

Therefore, we have;

2 nonreal complex roots

2) Give that the graph of the function touches the x-axis once, we have;

1 Real Solution

3) The given function is f(x) = 2·x² + 8·x + 6

The general form of the quadratic function is f(x) = a·x² + b·x + c

Comparing, we have;

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The discriminant of the function, D = b² - 4·a·c, therefore, for the function, we have;

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The discriminant of the function, D = 16

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A vertical translation is given by the following equation;

y = f(x) + b

A horizontal to the right by 'a' translation is given by an equation of the form; y = f(x - a)

A vertical reflection is given by an equation of the form; y = -f(x) = -x²

A narrowing is given by an equation of the form; y = b·f(x), where b < 1

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Now this situation began to dismember in 1877, when federal troops left the southern states and Reconstruction officially came to an end. From then on, Democratic governors and legislators began to sanction the Black Codes and Jim Crow Laws, aimed at curtailing the civil and political rights of African-Americans.

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