the height of a ball is modelled by the equation h(t)= 4sin (8 pie t)+ 6.5 where h(t) is in metres and t is in seconds. what are the highest and lowest points the ball reaches?

a) 10.5 m and 6.5m
b) 10.5m and 2.5m
c) 6.5m and 2.5m
d) 14.5m and 6.5 m

answer is C, you can graph it, and see the extremas

Well, well, well, looks like the ball is reaching for the stars...or maybe just the highest point. Let's crunch some numbers and find out!

To find the highest and lowest points the ball reaches, we need to look at the amplitude of the sin function. In this case, the amplitude is 4. So, the highest point the ball reaches will be 4 units above the midpoint, which is 6.5m.

So, the highest point is 6.5 + 4 = 10.5m. Is this the right answer? Time will tell!

Now, for the lowest point, we need to consider that the sin function also dips below the midpoint. Since the amplitude is 4, the lowest point will be 4 units below the midpoint, which is 6.5m.

So, the lowest point is 6.5 - 4 = 2.5m. Are you ready for it? That's the answer!

Therefore, the highest and lowest points the ball reaches are 10.5m and 2.5m, respectively. So, the correct answer is b) 10.5m and 2.5m. Keep bouncing those math questions at me!

To find the highest and lowest points the ball reaches, we need to determine the maximum and minimum values of the function h(t) = 4sin(8πt) + 6.5.

The maximum and minimum values of a sine function occur when the sine value is at its highest and lowest points, respectively.

The maximum value of the sine function is 1, and the minimum value is -1. Therefore, the maximum and minimum values of the function h(t) occur when the sine term is 1 and -1, respectively.

For the maximum height, we set the sine term equal to 1 and solve for t:

4sin(8πt) = 1

Dividing both sides by 4:

sin(8πt) = 1/4

To find the smallest positive value of t that satisfies this equation, we use the inverse sine function (sin⁻¹):

8πt = sin⁻¹(1/4)

Using a calculator, the smallest positive value of t that satisfies this equation is approximately 0.0727 seconds.

Now, let's find the maximum height by substituting this value of t into the equation h(t):

h(0.0727) = 4sin(8π * 0.0727) + 6.5 = 10.5 meters

Therefore, the highest point the ball reaches is 10.5 meters.

For the minimum height, we set the sine term equal to -1 and solve for t:

4sin(8πt) = -1

Dividing both sides by 4:

sin(8πt) = -1/4

To find the smallest positive value of t that satisfies this equation, we use the inverse sine function:

8πt = sin⁻¹(-1/4)

Using a calculator, the smallest positive value of t that satisfies this equation is approximately 0.2356 seconds.

Now, let's find the minimum height by substituting this value of t into the equation h(t):

h(0.2356) = 4sin(8π * 0.2356) + 6.5 = 2.5 meters

Therefore, the lowest point the ball reaches is 2.5 meters.

In conclusion, the highest and lowest points the ball reaches are 10.5 meters and 2.5 meters, respectively. Therefore, the correct answer is option b) 10.5m and 2.5m.

To find the highest and lowest points the ball reaches, we need to identify the maximum and minimum values of the function h(t) = 4sin(8πt) + 6.5.

To do this, we can analyze the amplitude (A) of the sine function. The amplitude represents the maximum distance the graph of the function reaches above and below its midline.

In the given equation, the coefficient of sin(8πt) is 4. This coefficient represents the amplitude. So, the amplitude of the function is |4| = 4.

Since the midline of the function is 6.5, we can add and subtract the amplitude from the midline to find the highest and lowest points the ball reaches.

Highest point = midline + amplitude = 6.5 + 4 = 10.5 meters
Lowest point = midline - amplitude = 6.5 - 4 = 2.5 meters

Therefore, the highest and lowest points the ball reaches are 10.5 meters and 2.5 meters respectively.

So, the correct answer is option b) 10.5m and 2.5m.