Question
A weight is attached to a spring that is fixed to the floor. The
equation h = 7 cos (Et)
models the height, h, in centimeters after t seconds of the weight being stretched and released.
a. Solve the equation for t.
b. Find the times at which the weight is first at a height of 1 cm, of 3 cm, and of 5 cm above the rest position.
Calculate your answer using radian angle measure. Round your answers to the nearest hundredth.
equation h = 7 cos (Et)
models the height, h, in centimeters after t seconds of the weight being stretched and released.
a. Solve the equation for t.
b. Find the times at which the weight is first at a height of 1 cm, of 3 cm, and of 5 cm above the rest position.
Calculate your answer using radian angle measure. Round your answers to the nearest hundredth.
Answers
GPT 3.5
a. To solve the equation for t, we need to isolate t.
h = 7 cos (Et)
Divide both sides by 7:
h/7 = cos (Et)
Inverse cosine both sides:
cos^(-1) (h/7) = Et
t = cos^(-1) (h/7) / E
b. To find the times at which the weight is first at a height of 1 cm, 3 cm, and 5 cm above the rest position, we substitute each height value into the equation and solve for t.
For h = 1 cm:
t = cos^(-1) (1/7) / E = 0.813 / E
For h = 3 cm:
t = cos^(-1) (3/7) / E = 1.231 / E
For h = 5 cm:
t = cos^(-1) (5/7) / E = 1.375 / E
Therefore, the weight is first at a height of 1 cm above the rest position at approximately 0.81 / E seconds, at a height of 3 cm above the rest position at approximately 1.23 / E seconds, and at a height of 5 cm above the rest position at approximately 1.38 / E seconds.
h = 7 cos (Et)
Divide both sides by 7:
h/7 = cos (Et)
Inverse cosine both sides:
cos^(-1) (h/7) = Et
t = cos^(-1) (h/7) / E
b. To find the times at which the weight is first at a height of 1 cm, 3 cm, and 5 cm above the rest position, we substitute each height value into the equation and solve for t.
For h = 1 cm:
t = cos^(-1) (1/7) / E = 0.813 / E
For h = 3 cm:
t = cos^(-1) (3/7) / E = 1.231 / E
For h = 5 cm:
t = cos^(-1) (5/7) / E = 1.375 / E
Therefore, the weight is first at a height of 1 cm above the rest position at approximately 0.81 / E seconds, at a height of 3 cm above the rest position at approximately 1.23 / E seconds, and at a height of 5 cm above the rest position at approximately 1.38 / E seconds.