What is the rate of change in the equation y=-2x+7

1. A buoy floats 19 yards from the eastern most point of a boat and 15 yards from the western most point of a second boat. The a

Black_prince [1.1K]

The laws of cosines and law of sines can be used given that two sides

and an included angle, or two angles a side are known.

Response:

1. The other angles in the triangle formed by the buoy are approximately;

31.1° and 40.9°

2. Distance of the helicopter from the first island is approximately;

14.5 miles

<h3>How is the Law of Sines and Cosines used?</h3>

Given parameters are;

Distance of the buoy from the easternmost point of a boat = 19 yards

Distance of the buoy from the westernmost point of the other boat = 15 yards

Angle formed from the buoy to the two boats = 108°

Distance between the two boats, d, is given by the law of cosines, as follows;

d² = 19² + 15² - 2 × 19 × 15 × cos(108°) = 586 - 570·cos(108°)

d = √(586 - 570·cos(108°))

By the law of Sines, we have;

$\dfrac{d}{sin(108^{\circ})} = \mathbf{\dfrac{15}{sin(Angle \ formed \ from \ the \ boat \ on \ the \ West, \ \theta_1)}}$

Which gives;

$sin(\theta_1) = \mathbf{ \dfrac{15 \times sin(108^{\circ})}{\sqrt{586 - 570 \cdot cos(108^{\circ})} }}$

The o

$\theta_1 = arcsin \left( \dfrac{15 \times sin(108^{\circ})}{\sqrt{586 - 570 \cdot cos(108^{\circ})} } \right) \approx \mathbf{31.1^{\circ}}$

The other angles formed in the triangle containing the buoy are;

θ₁ ≈ 31.1

θ₂ ≈ 180° - 108° - 31.1° ≈ 40.9°

2. Distance between the two islands = 20 miles

Angle of elevation with one island = 15°

Angle of elevation with the second island = 35°

Required:

The mileage (distance travelled) of the helicopter.

Solution:

Let A represent the island that has an angle of elevation to the helicopter

of 15°, and let B represent the other island.

Angle formed by the helicopter and the two island, θ, is found as follows;

θ = 180° - (15° + 35°) = 130°

By the Law of Sines, we have;

$\dfrac{20}{sin(130^{\circ})} = \mathbf{ \dfrac{Distance \ from \ island \ A }{sin(35^{\circ})}}$

Which gives;

$Distance \ of \ helicopter \ from \ island \ A = \mathbf{ \dfrac{20}{sin(130^{\circ})} \times sin(35^{\circ})}$

$Mileage \ from \ island \ A = \dfrac{20}{sin(130^{\circ})} \times sin(35^{\circ}) \times cos(15^{\circ}) \approx 14.5$

The mileage of the helicopter from the first island is approximately 14.5 miles

Learn more about the Law of Sines and Cosines here:

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7 0

2 years ago

The height of water shooting from a fountain is modeled by the function f(x) = −0.2x2 − 2.8x − 5.4 where x is the distance from

balandron [24]

Start by setting the function equal to 0 and then moving the -5.4 over to the other side of the equals sign. $-.2 x^{2} -2.8x=5.4$ . The first rule for completing the square is that the leading coefficient be a +1. Ours is a -.2. So we need to factor it out. $-.2( x^{2} +14x)=5.4$ . Now we will take half the linear term, square it, and add it to both sides. Our linear term is 14. Half of 14 is 7, and 7 squared is 49. So we add 49 in to the left side just fine, but we cannot forget about that -.2 hanging around out front as a multiplier. What we have actually "added" in is -.2*49 which is -9.8. Now here's what we have after all that: $-.2( x^{2} +14x+49)=5.4-9.8$ . In that process, we have created a perfect square binomial on the left. Along with expressing that binomial we will do the math on the right: $-.2(x+7) ^{2} =-4.4$ . Now we will move the -4.4 back over by addition, and it will then be apparent as to what our vertex is. The y coordinate of the vertex will give us the max height of the water. $-.2(x+7) +4.4=y$ . As you can see, our work matches choice C from above.

4 0

3 years ago

Really easy math question

Eddi Din [679]

Number of sodas sold = 130

Number of hot dogs sold = 65

Step-by-step explanation:

Given,

Combined number of sodas and hot dogs sold = 195

Let,

x represent the number of sodas sold

y represent the number of hot dogs sold