Jill wants to have a bowling party with her friends. Normally, the local bowling alley charges \( \$ 4 \) per person to bowl and \( \$ 3 \) per person for shoe rentals. However, the alley also has a group rate of \( \$ 90 \) for everyone to bowl, along with a discounted rate of \( \$ 2 \) per person for shoe rentals. How many people would need to go to the party for the two options to cost the same? \( \square \) people

\( 1 \leftarrow \quad \) Fill in the blanks to complete the fundamental identities. \[ \tan \theta=\frac{1}{\square}=\frac{\sin \theta}{\square} \]

Assume the cost of a gallon of milk is \( \$ 3.50 \). With continuous compounding, find the time it would take the cost to be 3 times as much (to the nearest tenth of a year), at an annual inflation rate of \( 6 \% \). A. 18.31 years

Perform the indicated operation and simplify the result. \( \cot x(\tan x-\sec x) \) The answer is

With the current housing market situation in the United States, iths estimated that the number of houses built, \( H(t) \) (in millions) over the next 5 years, is related to the mortgage rate, \( r(t) \) (percent per year), and can be modeled by the equation: \[ 7 H(t)^{2}+r(t)=32 \] Find the rate of change of the number of houses being built with respect to time when the mortgage rate is \( 8 \% \) per year and is increasing at \( 0.1 \% \) per year. (Round your answer to the nearest whole number.)

\( 1 \leftarrow \quad \) Fill in the blank to complete the fundamental identity. \[ \tan ^{2} \theta+1=\square \]