Angelique is thinking about buying a house for $179,000. the table below shows the projected value of two different houses for three years. number of years 1 2 3 house 1 (value in dollars) 186,160 193,606.40 201,350.66 house 2 (value in dollars) 190,000 201,000 212,000 part a: what type of function, linear or exponential, can be used to describe the value of each of the houses after a fixed number of years? explain your answer. (2 points) part b: write one function for each house to describe the value of the house f(x), in dollars, after x years. (4 points) part c: angelique wants to purchase a house that would have the greatest value in 30 years. will there be any significant difference in the value of either house after 30 years? explain your answer, and show the value of each house after 30 years. (4 points)

Part a: let's consider the house price as consecutive terms in a sequence.a linear sequence is determined if each consecutive term has the same difference.an exponential sequence is determined if each consecutive term has the same ratio.house a: term 1 = 186160term 2 = 193606.40term 3 = 201350.66we first check whether the function is a linear oneterm 2 - term 1 = 193606.40 - 186160 = 7446.4term 3 - term 2 = 201350.66 - 193606.40 = 7744.26the function isn't linear. it doesn't increase in the same constant.we check if the sequence has the same ratioterm 2/term 1 = 193606.40 / 186160 = 1.04term 3/term 2 = 201350.66 / 193606.40 = 1.04the consecutive terms have the same ratio; the sequence is we do the same check with house b's price increaseterm 1 = 190000term 2 = 201000term 3 = 212000finding the differenceterm 2 - term 1 = 201000 - 190000 = 11000term 3 - term 2 = 212000 - 201000 = 11000the sequence has the same difference between each consecutive term. hence the sequence is part b: house a: the general form of an exponential function is f(x) = c(rˣ), wheref(x) = the final value after x yearsc = 179000r = 1.04x = the number of yearsthe function, f(x), for house a is f(x) = 179000 (1.04ˣ)house b: the general form of a  linear function is h(x) = mx + cwhereh(x) is the price of the house after x yearsm = the rate of changec = the initial valuethe function for house b is h(x) = 11000x + 179000part c: the value of house a after 30 yearsf(30) = 179000 (1.04)³⁰ = 580568.15the value of house b after 30 yearsf(30) = 11000(30) + 179000 = 509000there is a significant difference between the value of house a and house b in 30 

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12