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3 years ago

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A rotating lawn sprinkler sprays water in a circular area of grass as shown in the picture. The diameter of the circular area of

grass is 6 feet. How many square feet of water will the sprinkler spray?

1 answer:

Georgia [21]3 years ago

7 0

Answer:

$9\pi \: \mathrm{ft^2\:or}\: 28.27 \mathrm{\: ft^2}$

Step-by-step explanation:

The area of a circle is given by $r^2\pi$ where $r$ is the radius of the circle. The diameter is given as $6$ feet, so the radius of this circle is $\frac{6}{2}=3$ feet.

Plugging in our values, we get:

$3^2\pi=9\pi\approx28.27$ .

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The value of $cos(a+b)$ for the angles $a$ and $b$ in standard position in the first quadrant is $-\frac{36}{85}$

We need to find the value of $cos(a+b)$ . To proceed, we need to use the compound angle formula

<h3>Cosine of a sum of two angles</h3>

The cosine of the sum of two angles $a$ and $b$ is given below

$cos(a+b)=cos(a)cos(b)-sin(a)sin(b)$

We are given

$sin(a)=\dfrac{15}{17}\\\\cos(b)=\dfrac{3}{5}$

We need to find $sin(b)$ and $cos(a)$ , using the identity

$sin^2(\theta)+cos^2(\theta)=1$

<h3>Find sin(b)</h3>

To find $sin(b)$ , note that

$sin^2(b)+cos^2(b)=1\\\\\implies sin(b)=\sqrt{1-cos^2(b)}$

substituting $\frac{3}{5}$ for $cos(b)$ in the identity, we get

$sin(b)=\sqrt{1-cos^2(b)}\\\\=\sqrt{1-\left(\dfrac{3}{5}\right)^2}=\dfrac{4}{5}$

<h3>Find cos(a)</h3>

To find $cos(a)$ , note that

$sin^2(a)+cos^2(a)=1\\\\\implies cos(a)=\sqrt{1-sin^2(a)}$

substituting $\frac{15}{17}$ for $sin(a)$ in the identity, we get

$cos(a)=\sqrt{1-sin^2(a)}\\\\=\sqrt{1-\left(\dfrac{15}{17}\right)^2}=\dfrac{8}{17}$

<h3>Find the value of cos(a+b)</h3>

We can now make use of the formula

$cos(a+b)=cos(a)cos(b)-sin(a)sin(b)$

to find $cos(a+b)$ .

$cos(a+b)=cos(a)cos(b)-sin(a)sin(b)\\\\=\dfrac{8}{17}\cdot\dfrac{3}{5}-\dfrac{15}{17}\cdot\dfrac{4}{5}=-\dfrac{36}{85}$

Learn more about sine and cosine of compound angles here brainly.com/question/24305408

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2 years ago