32 X (-5)

Looking at the top of tower A and base of tower B from points C and D, we find that ∠ACD = 60°, ∠ADC = 75° and ∠ADB = 30°. Let t

katrin2010 [14]

Answer:

$\text{Exact: }AB=25\sqrt{6},\\\text{Rounded: }AB\approx 61.24$

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:

$\angle CAD+\angle ACD+\angle CDA=180^{\circ},\\\angle CAD+60^{\circ}+75^{\circ}=180^{\circ},\\\angle CAD=180^{\circ}-75^{\circ}-60^{\circ},\\\angle CAD=45^{\circ}$

Now use this value in the Law of Sines to find AD:

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

Recall that $\sin 45^{\circ}=\frac{\sqrt{2}}{2}$ and $\sin 60^{\circ}=\frac{\sqrt{3}}{2}$ :

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

Now that we have the length of AD, we can find the length of AB. The right triangle $\triangle ADB$ is a 30-60-90 triangle. In all 30-60-90 triangles, the side lengths are in the ratio $x:x\sqrt{3}:2x$ , where $x$ is the side opposite to the 30 degree angle and $2x$ is the length of the hypotenuse.

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:

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

3 0

3 years ago

Read 2 more answers

Will give brainless if solved

wel

Answer:

1. 70 2. 85 3. 63 4. 20 5. They are vertical angles

Step-by-step explanation:

6 0

3 years ago

Angle ABC is shown . Determine sin B. Write your answer in fraction form.<br><br> Help ASAP

Mkey [24]

Answer:

sin b would be 4/5

Step-by-step explanation:

sine is the opposite leg/the hypotenuse

7 0

2 years ago

A rectangular sandbox has a width of 5 feet.The sandbox is 5 times as long as it is wide. What is the perimeter of the sandbox?

AveGali [126]

Wide is 5 x 5 = 25 feet;
P = 2 x width + 2 x wide = 10 + 50 = 60 feet;

8 0

3 years ago

A TV that usually sells for $192.74 is on sale for 25% off. If sales tax on the TV is 5%, what is the price of the TV, including

Mashutka [201]

Answer:

Step-by-step explanation:

If you are saving 25%, you are still paying 75%, right? So the cost of the TV with a discount of 25% is 192.74(.75) which is $144.56

That's the cost of the TV on sale. Now we need to find the sales tax amount, if the sales tax is .05 as a decimal. We are paying $144.56 for the TV, which is 100% of the cost. On top of that, we are paying another 5%. So altogether, for the TV, we are paying 105% of the TV. Therefore,

144.56(1.05) = $151.79

4 0

3 years ago