Kajal holds a ruler in some wavy water. The depth of the water t seconds after she starts measuring it, in cm, is given by D(t)=50-23 sin(pi(t+0.23)) After she starts measuring, when is the first time the depth of the waves is at its average value? Give an exact answer. When t= ◻ seconds

Find Average Depth: To find the average value of the wave depth, we need to determine the constant term in the equation D(t) = 50 - 23 sin(pi(t+0.23)). The average depth is the constant term since the average value of the sine function over its period is zero. Calculation: Average depth = 50 cm.Set Depth to Average: Next, we need to find when the depth equals the average depth. Set D(t) to 50: 50 = 50 - 23 sin(pi(t+0.23)). Calculation: 0 = -23 sin(pi(t+0.23)).Solve for Sine Function: Solve for sin(pi(t+0.23)) = 0. The sine function equals zero at integer multiples of pi. Calculation: pi(t+0.23) = n*pi, where n is an integer.Calculate t Value: Solve for t: t + 0.23 = n. Calculation: t = n - 0.23.Determine Smallest Positive t: The smallest positive value for t occurs when n = 1 (since n = 0 gives a negative time, which doesn't make sense in this context). Calculation: t = 1 - 0.23 = 0.77 seconds.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Kajal holds a ruler in some wavy water. The depth of the water
t seconds after she starts measuring it, in
cm, is given by

D(t)=50-23 sin(pi(t+0.23))
After she starts measuring, when is the first time the depth of the waves is at its average value? Give an exact answer.
When
t=
◻ seconds

Kajal holds a ruler in some wavy water. The depth of the water $\newline$ $t$ seconds after she starts measuring it, in $\newline$ $\text{cm}$ , is given by $\newline$ $D(t)=50-23 \sin(\pi(t+0.23))$ $\newline$ After she starts measuring, when is the first time the depth of the waves is at its average value? Give an exact answer. $\newline$ When $\newline$ $t=$ $\newline$ $\square$ seconds

View step-by-step help

Home

Math Problems

Grade 7

Evaluate two-variable equations: word problems

Full solution

Q. Kajal holds a ruler in some wavy water. The depth of the water

\newline

t

seconds after she starts measuring it, in

\newline

\text{cm}

, is given by

\newline

D(t)=50-23 \sin(\pi(t+0.23))

\newline

After she starts measuring, when is the first time the depth of the waves is at its average value? Give an exact answer.

\newline

When

\newline

t=

\newline

\square

seconds

Find Average Depth: To find the average value of the wave depth, we need to determine the constant term in the equation $D(t) = 50 - 23 \sin(\pi(t+0.23))$ . The average depth is the constant term since the average value of the sine function over its period is zero. $\newline$ Calculation: Average depth = $50$ cm.
Set Depth to Average: Next, we need to find when the depth equals the average depth. Set $D(t)$ to $50$ : $\newline$ $50 = 50 - 23 \sin(\pi(t+0.23)).$ $\newline$ Calculation: $0 = -23 \sin(\pi(t+0.23)).$
Solve for Sine Function: Solve for $\sin(\pi(t+0.23)) = 0$ . The sine function equals zero at integer multiples of $\pi$ . $\newline$ Calculation: $\pi(t+0.23) = n\pi$ , where $n$ is an integer.
Calculate t Value: Solve for $t$ : $t + 0.23 = n$ . $\newline$ Calculation: $t = n - 0.23$ .
Determine Smallest Positive $t$ : The smallest positive value for $t$ occurs when $n = 1$ (since $n = 0$ gives a negative time, which doesn't make sense in this context). $\newline$ Calculation: $t = 1 - 0.23 = 0.77$ seconds.