Question

questions 22 and 23 refer to the following information. annual budgets for different programs in kansas, 2007 to 2010
program year
2007 2008 2009 2010
agriculture/natural resources 373,904 358,708 485,807 488,106
education 2,164,607 2,413,984 2,274,514 3,008,036
general government 14,347,325 12,554,845 10,392,107 14,716,155
highways and transportation 1,468,482 1,665,636 1,539,480 1,773,893
human resources 4,051,050 4,099,067 4,618,444 5,921,379
public safety 263,463 398,326 355,935 464,233
the table above lists the annual budget, in thousands of dollars, for each of six different state programs in kansas from 2007 to 2010.
22 which of the following best approximates the average rate of change in the annual budget for agriculture/natural resources in kansas from 2008 to 2010?
a) $50,000,000 per year
b) $65,000,000 per year
c) $75,000,000 per year
d) $130,000,000 per year
23 of the following, which programs ratio of its 2007 budget to its 2010 budget is closest to the human resources programs ratio of its 2007 budget to its 2010 budget?
a) agriculture/natural resources
b) education
c) highways and transportation
d) public safety
24 which of the following is an equation of a circle in the xy - plane with center (0, 4) and a radius with endpoint (4/3,5)?
a) $x^{2}+(y - 4)^{2}=\frac{25}{9}$
b) $x^{2}+(y + 4)^{2}=\frac{25}{9}$
c) $x^{2}+(y - 4)^{2}=\frac{5}{3}$
d) $x^{2}+(y + 4)^{2}=\frac{3}{5}$
25 $h=-4.9t^{2}+25t$
the equation above expresses the approximate height h, in meters, of a ball t seconds after it is launched vertically upward from the ground with an initial velocity of 25 meters per second. after approximately how many seconds will the ball hit the ground?
a) 3.5
b) 4.0
c) 4.5
d) 5.0
26 katarina is a botanist studying the production of pears by two types of pear trees. she noticed that type a trees produced 20 percent more pears than type b trees did. based on katarinas observation, if the type a trees produced 144 pears, how many pears did the type b trees produce?
a) 115
b) 120
c) 124
d) 173
27 a square field measures 10 meters by 10 meters. ten students each mark off a randomly selected region of the field; each region is square and has side lengths of 1 meter, and no two regions overlap. the students count the earthworms contained in the soil to a depth of 5 centimeters beneath the grounds surface in each region. the results are shown in the table below.
region number of earthworms region number of earthworms
a 107 f 141
b 147 g 150
c 146 h 154
d 135 i 176
e 149 j 166
which of the following is a reasonable approximation of the number of earthworms to a depth of 5 centimeters beneath the grounds surface in the entire field?
a) 150
b) 1,500
c) 15,000
d) 150,000

Explanation:

22

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x_1$ to $x_2$ is $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$. Here, $x_1 = 2008$, $x_2=2010$, and $f(x)$ is the budget for agriculture/natural resources. The budget in 2008 is $358708$ thousand dollars and in 2010 is $488106$ thousand dollars.

Step2: Calculate the average rate of change

The average rate of change $=\frac{488106 - 358708}{2010 - 2008}=\frac{129398}{2}=64699$ thousand dollars per year. Since $1$ thousand dollars = $1000$ dollars, $64699$ thousand dollars per year $\approx65000000$ dollars per year.

Step1: Calculate the ratio for agriculture/natural resources

The ratio of 2007 budget to 2010 budget for agriculture/natural resources is $\frac{373904}{488106}\approx0.766$.

Step2: Calculate the ratio for education

The ratio of 2007 budget to 2010 budget for education is $\frac{2164607}{3008036}\approx0.719$.

Step3: Calculate the ratio for highways and transportation

The ratio of 2007 budget to 2010 budget for highways and transportation is $\frac{1468482}{1773893}\approx0.828$.

Step4: Calculate the ratio for public safety

The ratio of 2007 budget to 2010 budget for public safety is $\frac{263463}{464233}\approx0.568$.

Step5: Calculate the ratio for human resources

The ratio of 2007 budget to 2010 budget for human resources is $\frac{4051050}{5921379}\approx0.684$.
We want to find which program's ratio is closest to the human - resources ratio. Comparing the differences:
For agriculture/natural resources: $|0.766 - 0.684|=0.082$.
For education: $|0.719 - 0.684| = 0.035$.
For highways and transportation: $|0.828-0.684|=0.144$.
For public safety: $|0.568 - 0.684|=0.116$.
The smallest difference is for education.

Step1: Recall the distance formula to find the radius

The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. The center of the circle is $(0,4)$ and a point on the circle is $(\frac{4}{3},5)$. The radius $r=\sqrt{(\frac{4}{3}-0)^2+(5 - 4)^2}=\sqrt{\frac{16}{9}+1}=\sqrt{\frac{16 + 9}{9}}=\sqrt{\frac{25}{9}}=\frac{5}{3}$.

Step2: Recall the standard form of the equation of a circle

The standard form of the equation of a circle with center $(h,k)$ and radius $r$ is $(x - h)^2+(y - k)^2=r^2$. Here, $h = 0$, $k = 4$ and $r=\frac{5}{3}$, so the equation is $x^{2}+(y - 4)^{2}=\frac{25}{9}$.

Answer:

B

23