A weight is attached to the end of a spring. Its height after $t$ seconds is given by the equation $\newline$ $h(t)=5-2 \sin \left(\frac{2 \pi(t+1)}{7}\right) .$ $\newline$ When does the weight first reach its maximum height? Give an exact answer. $\newline$ When $t= \square$ seconds

Full solution

Q. A weight is attached to the end of a spring. Its height after

t

seconds is given by the equation

\newline

h(t)=5-2 \sin \left(\frac{2 \pi(t+1)}{7}\right) .

\newline

When does the weight first reach its maximum height? Give an exact answer.

\newline

When

t= \square

seconds

Find Maximum Height: The maximum height is reached when the sine function is at its maximum value, which is $1$ .
Find Value of t: We need to find the value of $t$ for which $\sin\left(\frac{2\pi(t+1)}{7}\right)$ equals $1$ .
Set Sine Function Equal: Set the inside of the sine function equal to $\pi/2$ , because $\sin(\pi/2) = 1$ .
Solve for $t$ : $\frac{2\pi(t+1)}{7} = \frac{\pi}{2}$ . Now solve for $t$ .
Multiply by $7$ : Multiply both sides by $7$ to get rid of the denominator: $2\pi(t+1) = \frac{7\pi}{2}$ .
Divide by $2$ $\pi$ : Divide both sides by $2$ $\pi$ to solve for $(t+1)$ : $t+1 = \frac{7\pi}{4\pi}$ .
Simplify Right Side: Simplify the right side: $t+1 = \frac{7}{4}$ .
Subtract $1$ : Subtract $1$ from both sides to solve for $t$ : $t = \frac{7}{4} - 1$ .
Convert to Common Denominator: Convert $1$ to $\frac{4}{4}$ to have a common denominator: $t = \frac{7}{4} - \frac{4}{4}.$
Add Fractions: Add the fractions: $t = \frac{7-4}{4}$ .
Calculate Result: Calculate the result: $t = \frac{3}{4}$ .