A weight attached to a spring is at its lowest point, 9 inches below equilibrium, at time \( t=0 \) seconds. When the weight it released, it oscillates and returns to its original position at \( t=3 \) seconds. Which of the following equations models the distance, \( d \), of the weight from its equilibrium after \( t \) seconds? \( d=-9 \cos \left(\frac{\pi}{3} t\right) \) \( d=-9 \cos \left(\frac{2 \pi}{3} t\right) \) \( d=-3 \cos \left(\frac{\pi}{9} t\right) \) \( d=-3 \cos \left(\frac{2 \pi}{9} t\right) \)

15. A city of 100,000 is having pollution problems and is decreasing in size \( 2 \% \) annually (every year). What is the domain and range for this sitoation?

-ind the standard form of the equation of the hyperbola satisfying the given conditions. -intercepts \( \pm 12 \); foci at \( (-13,0) \) and \( (13,0) \) The equation in standard form of the hyperbola is \( \square \). (Simplify your answer. Use integers or fractions for any numbers in the equation.)

Graph the given equation. Give the domain and range. Identify any that are graphs of functions. \( \frac{y}{2}=\sqrt{1-\frac{x^{2}}{49}} \) Graph \( \frac{y}{2}=\sqrt{1-\frac{x^{2}}{49}} \). The domain of the the equation \( \frac{y}{2}=\sqrt{1-\frac{x^{2}}{49}} \) is (Type your answer in interval notation.) The range of the the equation \( \frac{y}{2}=\sqrt{1-\frac{x^{2}}{49}} \) is (Type your answer in interval notation.) Does the given equation define the graph of a function? Yes No

A laboratory designed a radio telescope that had a diameter of 300 ft and maximum depth of 46 ft . (a)Write an equation of a parabola that models the cross section of the dish if the vertex is placed at the origin and the parabola opens up. (b) The receiver must be placed at the focus of the parabola. How far from the vertex should the receiver be located? (a) The equation of the parabola is \( \square \). (Simplify your answer. Use integers or fractions for any numbers in the equation.) (b) The receiver should be located at approximately \( \square \mathrm{ft} \) above the vertex. (Type an integer or decimal rounded to the nearest tenth as needed.)

For the exponential function \( y=3(1.6)^{x} \), complete parts (a) through (g). (a) Does the function represent exponential growth or decay? A. The function represents exponential decay.