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2 years ago

1/2r -3= 3 (4-3/2r) solve for r

Mathematics

1 answer:

s2008m [1.1K]2 years ago

8 0

1/2r -3= 3 (4-3/2r) is to be solved for r.

I'll begin by making the assumptions that by 1/2r you actually meant (1/2)r and that by 3/2r you actually meant (3/2)r. When in doubt, please use parentheses to make your meaning clear.

Thus, 1/2r -3= 3 (4-3/2r) becomes (1/2)r -3= 3 (4-(3/2)r ) .

Simplify this by multiplying all 3 terms by 2. Doing this will eliminate the fractions:

r -6 = 3 (4*2-(3)r ) or r - 6 = 24 - 9r

Now expand the right side, using the distributive property of

r - 3 = 24 - 9r

Regrouping so as to combine like terms:

10r = 30

Solving for r: r = 30/10 = 3

The value of r that satisfies this equation is 3

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Step-by-step explanation:

We can use the Law of Sines to find segment AD, which happens to be a leg of $\triangle ACD$ and the hypotenuse of $\triangle ADB$ .

The Law of Sines states that the ratio of any angle of a triangle and its opposite side is maintained through the triangle:

$\frac{a}{\sin \alpha}=\frac{b}{\sin \beta}=\frac{c}{\sin \gamma}$

Since we're given the length of CD, we want to find the measure of the angle opposite to CD, which is $\angle CAD$ . The sum of the interior angles in a triangle is equal to 180 degrees. Thus, we have:

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$\frac{AD}{\sin 60^{\circ}}=\frac{100}{\sin 45^{\circ}},\\\\AD=\sin 60^{\circ}\cdot \frac{100}{\sin 45^{\circ}}$

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$AD=\frac{\frac{\sqrt{3}}{2}\cdot 100}{\frac{\sqrt{2}}{2}},\\\\AD=\frac{50\sqrt{3}}{\frac{\sqrt{2}}{2}},\\\\AD=50\sqrt{3}\cdot \frac{2}{\sqrt{2}},\\\\AD=\frac{100\sqrt{3}}{\sqrt{2}}\cdot\frac{ \sqrt{2}}{\sqrt{2}}=\frac{100\sqrt{6}}{2}={50\sqrt{6}}$

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Since AD is the hypotenuse, it must represent $2x$ in this ratio and since AB is the side opposite to the 30 degree angle, it must represent $x$ in this ratio (Derive from basic trig for a right triangle and $\sin 30^{\circ}=\frac{1}{2}$ ).

Therefore, AB must be exactly half of AD:

$AB=\frac{1}{2}AD,\\AB=\frac{1}{2}\cdot 50\sqrt{6},\\AB=\frac{50\sqrt{6}}{2}=\boxed{25\sqrt{6}}\approx 61.24$

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