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2 years ago

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Triangle Z W X is shown. Angle Z W X is a right angle. An altitude is drawn from point W to point Y on side Z X to form a right

angle. The length of W X is b, the length of X Y is a, the length of Y Z is 5, and the length of W Y is 6. What is the approximate value of b, rounded to the nearest tenth? 7.2 units 7.8 units 8.6 units 9.4 units

1 answer:

sashaice [31]2 years ago

3 0

Answer:

b = 9.1units

Step-by-step explanation:

Find the diagram attached

Using the similarity theorem of triangle

WX/XZ = WY/WX

Substitute;

b/a+5 = 6/b

b² = 6(a+5) ... 1

Also according to pythagoras theorem on triangle WYX;

b² = 6²+a²...2

Equate 1 and 2

6(a+5) = 6²+a²

6a+30 = 36 + a²

a²-6a+36-30 = 0

a²-6a+6 = 0

a = 6±√36+4(6)/2

a = 6±√60/2

a = 6±7.745/2

a = 13.745/2

a = 6.8725

Recall that

b² = 6²+a²

b² = 36+6.8725²

b² = 83.2313

b = 9.12units

b = 9.1units

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The point P(7, −2) lies on the curve y = 2/(6 − x). (a) If Q is the point (x, 2/(6 − x)), use your calculator to find the slope

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Answer:

a) (i) $m = 2.22$ , (ii) $m = 2$ , (iii) $m = 2$ , (iv) $m = 2$ , (v) $m = 1.82$ , (vi) $m = 2$ , (vii) $m = 2$ , (viii) $m = 2$ ; b) $m \approx 2$ ; c) The equation of the tangent line to curve at P (7, -2) is $y = 2\cdot x + 12$ .

Step-by-step explanation:

a) The slope of the secant line PQ is represented by the following definition of slope:

$m = \frac{\Delta y}{\Delta x} = \frac{y_{Q}-y_{P}}{x_{Q}-x_{P}}$

(i) $x_{Q} = 6.9$ :

$y_{Q} =\frac{2}{6-6.9}$

$y_{Q} = -2.222$

$m = \frac{-2.222 + 2}{6.9-7}$

$m = 2.22$

(ii) $x_{Q} = 6.99$

$y_{Q} =\frac{2}{6-6.99}$

$y_{Q} = -2.020$

$m = \frac{-2.020 + 2}{6.99-7}$

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(iii) $x_{Q} = 6.999$

$y_{Q} =\frac{2}{6-6.999}$

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(iv) $x_{Q} = 6.9999$

$y_{Q} =\frac{2}{6-6.9999}$

$y_{Q} = -2.0002$

$m = \frac{-2.0002 + 2}{6.9999-7}$

$m = 2$

(v) $x_{Q} = 7.1$

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$y_{Q} = -1.980$

$m = \frac{-1.980 + 2}{7.01-7}$

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$y_{Q} =\frac{2}{6-7.001}$

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$m = \frac{-1.998 + 2}{7.001-7}$

$m = 2$

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$m \approx \frac{f(6.9999)+f(7.0001)}{2}$

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$m \approx 2$

The slope of the tangent line to the curve at P(7, -2) is 2.

c) The equation of the tangent line is a first-order polynomial with the following characteristics:

$y = m\cdot x + b$

Where:

$x$ - Independent variable.

$y$ - Depedent variable.

$m$ - Slope.

$b$ - x-Intercept.

The slope was found in point (b) (m = 2). Besides, the point of tangency (7,-2) is known and value of x-Intercept can be obtained after clearing the respective variable:

$-2 = 2 \cdot 7 + b$

$b = -2 + 14$

$b = 12$

The equation of the tangent line to curve at P (7, -2) is $y = 2\cdot x + 12$ .

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