Math (Calculator OK) #1

Let g(x) = 8x - 5. Which of the following is equivalent to g(g(x))?

64x - 45 To calculate g(g(x)), input g(x) into function g as shown below. g(g(x)) = 8(g(x)) - 5 Since g(x) = 8x - 5, the above equation becomes: g(g(x)) = 8(g(x)) - 5 = 8(8x - 5) - 5 = 64x - 40 - 5 64x - 45 so g(g(x)) = 64x - 45.

A circle in the xy-plane has a diameter with endpoints at (0,3) and (-4,0). Which of the following is an equation of the circle?

(x + 2)^2 + (y - 3/2)^2 = 25/4 The formula for a circle with a center at (h, k) and a radius of r is: (x - h)^2 + (y - k)^2 = r^2 Let's begin by finding the center. The center will be at the midpoint of the diameter. The h-coordinate will be the midpoint of the x-coordinates. h = 0 + (-4) / 2 = -2 The k-coordinate will be the midpoint of the y-coordinates. k = 3 + 0 / 2 = 3/2 Therefore, the center is ( -2, 3/2). Next, let's find the diameter d. Using the distance formula, we have: d^2 = (0 - (-4))^2 + (3 - 0)^2 = (4)^2 + (3)^2 = 16 + 9 = 25 Therefore: r^2 = (d/2)^2 = d^2 / 4 25 / 4 Let's put the equation into standard form. (x - (-2))^2 + (y - 3/2)^2 = 25/4 (x + 2)^2 + ( y - 3/2)^2 = 25/4 The correct answer is: (x + 2)^2 + ( y - 3/2)^2 = 25/4

Which of the following is equivalent to: (5x^2 - 6x + 7) - (3x^3 + 4x^2 - 8)?

-3x^3 + x^2 - 6x + 15 Let's subtract the two polynomials with (5x^2 - 6x + 7) on top, as it came first in the problem. 5x^2 - 6x + 7 -(3x^3 + 4x^2 - 8) We'll put in some 0 placeholders so it is evident which like terms will combine. 0x^3 + 5x^2 - 6x + 7 -(3x^3 - 4x^2 - 0x + 8) Now we can distribute the negative sign. 0x^3 + 5x^2 - 6x + 7 + (-3x^3 - 4x^2 - 0x + 8) Now let's add. The final answer is: -3x^3 + x^2 - 6x + 15

At the beginning of January 2002, the price of ground beef was $1.70 per pound and the price of tuna fish was $2.20 per pound. For the following 15 months, the price of ground beef increased at the rate of $0.03 per month and the price of tuna fish decreased at $0.02 per month. In approximately how many months after the beginning of January 2002 was the price of ground beef and tuna fish the same, and what was the price?

10 months and $2.00 Develop an equation for the price for each item, x months in the future from January, 2002. Ground beef started at $1.70 per pound and increased $0.03 each month. The price x months in the future is: 1.70 + 0.03x Tuna fish started at $2.20 per pound and decreased $0.02 each month. The price x months in the future is: 2.20 - 0.02x Setting these expressions equal to one another will allow us to determine in how many months the price is the same: 1.70 + 0.03x = 2.20 - 0.02x 0.03x + 0.02x = 2.20 - 1.70 0.05x = 0.50 x = 10 Therefore, 10 months after January, 2002 the price of ground beef and tuna fish were the same. To determine the price at that time, 10 can be substituted into one of the expressions for the price: 1.70 + 0.03x 1.70 + 0.03x * 10 1.70 + 0.30 2.00 The price in 10 months for both ground beef and tuna fish was $2.00.

A chef has a large container of olive oil. In one night, after he used 25 quarts, 35.9% of the oil remained. How many quarts of olive oil remained in the container?

14 If 35.9% of the oil remained after the chef used it, then we can say that the chef used (100 - 35.9)% which is 64.1%, of the oil. Let's use the variable q to represent the total number of quarts of oil. We can then say: 0.641q = 25 The total number of quarts, then, after dividing by 0.641, is 39 quarts of oil. Now that we know the total, we can subtract 25 from 39 to find out how many quarts of oil remained. 39 - 25 = 14 12 quarts of oil remained in the container.

The scatterplot to the left depicts one study's projection of the world population, aged 808080 and above, from 195019501950 to 2050. A graph of an exponential function that models the data is shown, where t represents the years since 1950 and n represents the world population, aged 80 and over, in millions. According to the model, which of the following best approximates the number of people in the world that will be at least 80 years old in 2030?

180 million. We can solve this problem graphically by using the best fit curve drawn on the scatter plot above. Since t is years since 1950, 2030 corresponds to t = 80. Using the graph, we can see that t=80 corresponds to n = 180. This means that in 2030, approximately 180 million people will be aged 80 and over. 180 million best approximates the number of people in the world that will be at least 80 years old in 2030.

The table at left partially summarizes the percent of residents in different counties who participate in performing arts at least once a month. The percentages have a range of 12.9%. What could be the percentage of county G residents who participate in performing arts at least once a month?

22/5 or 88/5 To calculate the range, we need to find the difference between the highest and lowest values. If county G did not have the highest or lowest percentage, then the range would be: 17.3−4.7≠12.9 Thus, we know that county G must have either the highest or lowest percentage of residents who participate in the performing arts at least once per month. If county G had the lowest percentage, we can set up and solve the following equation. 17.3 − G = 12.9 17.3 = 12.9 + G 4.4 = G If county G had the highest percentage, we can set up and solve the following equation. G - 4.7 = 12.9 G = 17.6 In county G, either 4.4% or 17.6%, percent of residents participated in the performing arts at least once per month.

If k is a rational constant not equal to 1, which of the following graphs represents the equation y + 5 = k(x + y) + 5?

Because all of its variables have first degree exponents, we know that the graph of our equation must be a line. Let's rewrite the equation y + 5 = k(x + y) + 5 in y = mx + b form to reveal its slope and y-intercept. y + 5 = k(x +y) +5 y = k(x + y) y = kx + ky y -ky = kx y(1 - k) = kx y = kx / 1 - k The graph of y = kx / 1 - k has a y-intercept of 0, which tells is that it must pass through the origin.

m(t2) - m(t1) = 15(t2 -t1) A chemist dissolves a large amount of sodium acetate in boiling water. Upon bringing the temperature down to 4 ∘ Celsius, the solution loses stability and crystals start to grow with increasing mass m(t) in grams (g), t seconds after crystallization. The equation above shows this relationship where t1 and t2 are any values of t such that t1 < t2. Which of the following correctly explains the growth of the crystal mass?

The crystal mas grows linearly by 15 g per second. m(t2) - m(t1)/ t2 -t1 = 15 The expression on the left is the definition of average rate of change between times t1 and t2. Because this expression holds for all possible values of t1 and t2, this means that the rate of change in m with respect to t is a constant 15 everywhere. Therefore, the growth is linear. We have that the rate of change in m with respect to t is 15. Because the units of m are grams and the units of t are seconds, it must be that the crystal mass grows by 15 g per second. The crystal mass grows linearly by 15 g per second.

Between 2008 and 2012, the revenue obtained from digital music albums downloads,r, in millions of dollars, in the United States increased by approximately 132 million dollars per year. In 2010, the digital music revenue in the U.S. was about 872 million dollars. If t represents years since 2008, which of the following best models the situation for 0 ≤ t ≤ 4?

r(t) = 132t + 608 Since the revenue obtained from digital music album downloads in the United States increased by approximately 132 million dollars per yer, we have a constant rate of change, and thus can find a linear function to model the situation. Recall that y - y1 = m(x - x1) models a linear function with slope m passing through the point (x1, y1). Since t = 0 corresponds to year 2008, and since in 2010, the digital music revenue in the US was about $872 million, we know that the ordered pair (2, 872) must fit our model. The rate of change is equal to the slope of the line, so here we have m = 132. Substituting this information into the above point-slope model gives us the following: r - 872 = 132(t - 2) We can solve for r as shown below. r - 872 = 132(t -2) r - 872 = 132t - 264 r = 132t + 608 The function r(t) - 132t + 608 best models the situation.