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3 years ago

10

Find the value of the variables in the simplest form

2 answers:

Papessa [141]3 years ago

6 0

Answer:

x = 15 $\sqrt{3}$ , y = 15

Step-by-step explanation:

Using the sine and cosine ratios in the right triangle and the exact values

sin60° = $\frac{\sqrt{3} }{2}$ , cos60° = $\frac{1}{2}$ , then

sin60° = $\frac{opposite}{hypotenuse}$ = $\frac{x}{30}$ = $\frac{\sqrt{3} }{2}$ ( cross- multiply )

2x = 30 $\sqrt{3}$ ( divide both sides by 2 )

x = 15 $\sqrt{3}$

---------------------------------------------------------

cos60° = $\frac{adjacent}{hypotenuse}$ = $\frac{y}{30}$ = $\frac{1}{2}$ ( cross- multiply )

2y = 30 ( divide both sides by 2 )

y = 15

777dan777 [17]3 years ago

5 0

Answer:

Step-by-step explanation:

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3 years ago

Find the point (,) on the curve =8 that is closest to the point (3,0). [To do this, first find the distance function between (,)

ELEN [110]

Question:

Find the point (,) on the curve $y = \sqrt x$ that is closest to the point (3,0).

[To do this, first find the distance function between (,) and (3,0) and minimize it.]

Answer:

$(x,y) = (\frac{5}{2},\frac{\sqrt{10}}{2}})$

Step-by-step explanation:

$y = \sqrt x$ can be represented as: $(x,y)$

Substitute $\sqrt x$ for $y$

$(x,y) = (x,\sqrt x)$

So, next:

Calculate the distance between $(x,\sqrt x)$ and $(3,0)$

Distance is calculated as:

$d = \sqrt{(x_1-x_2)^2 + (y_1 - y_2)^2}$

So:

$d = \sqrt{(x-3)^2 + (\sqrt x - 0)^2}$

$d = \sqrt{(x-3)^2 + (\sqrt x)^2}$

Evaluate all exponents

$d = \sqrt{x^2 - 6x +9 + x}$

Rewrite as:

$d = \sqrt{x^2 + x- 6x +9 }$

$d = \sqrt{x^2 - 5x +9 }$

Differentiate using chain rule:

Let

$u = x^2 - 5x +9$

$\frac{du}{dx} = 2x - 5$

So:

$d = \sqrt u$

$d = u^\frac{1}{2}$

$\frac{dd}{du} = \frac{1}{2}u^{-\frac{1}{2}}$

Chain Rule:

$d' = \frac{du}{dx} * \frac{dd}{du}$

$d' = (2x-5) * \frac{1}{2}u^{-\frac{1}{2}}$

$d' = (2x - 5) * \frac{1}{2u^{\frac{1}{2}}}$

$d' = \frac{2x - 5}{2\sqrt u}$

Substitute: $u = x^2 - 5x +9$

$d' = \frac{2x - 5}{2\sqrt{x^2 - 5x + 9}}$

Next, is to minimize (by equating d' to 0)

$\frac{2x - 5}{2\sqrt{x^2 - 5x + 9}} = 0$

Cross Multiply

$2x - 5 = 0$

Solve for x

$2x =5$

$x = \frac{5}{2}$

Substitute $x = \frac{5}{2}$ in $y = \sqrt x$

$y = \sqrt{\frac{5}{2}}$

Split

$y = \frac{\sqrt 5}{\sqrt 2}$

Rationalize

$y = \frac{\sqrt 5}{\sqrt 2} * \frac{\sqrt 2}{\sqrt 2}$

$y = \frac{\sqrt {10}}{\sqrt 4}$

$y = \frac{\sqrt {10}}{2}$

Hence:

$(x,y) = (\frac{5}{2},\frac{\sqrt{10}}{2}})$

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3 years ago

A farmer ordered 3/4 ton of soil. He wants to divide it evenly into 8 flower beds. How much will he have for each bed?

nasty-shy [4]

We know that
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</span><span>divide it by 8
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the answer is
</span>93.75 kg f<span>or each bed</span>

7 0

3 years ago

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katen-ka-za [31]

Answer:

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Step-by-step explanation:

Let there are x psychology textbooks sold.

No of physics textbooks sold = 3x

ATQ,

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x + 3x = 392

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2 years ago