2015 Vocational Math Meet: Round 2 ALGEBRA 1 GEOMETRY ALGEBRA 2 RELATED TECHNICAL 200 200 200 200 400 400 400 400 600 600 600 600

Given the following system of equations 9x – 6y = 21 6x – 4y = k Algebra 1 200 points Given the following system of equations 9x – 6y = 21 6x – 4y = k find the value of k such that the system has an infinite number of solutions.

Algebra 1 200 points Answer Divide both sides of the first equation by 3 and both sides of the second equation by 2. 3x – 2y = 7 3x – 2y = 0.5k 0.5k =7 k = 14

Algebra 1 100 points Algebra 1 400 points The number 121 is the only perfect square that can be written as the sum of consecutive powers beginning with 1. Find a such that a0 + a1 + a2 + a3 + a4 = 121

a = 3 a0 = 1; so a1 + a2 + a3 + a4 = 120 44 = 256; so a ≠ 4 Algebra 1 400 points Answer a0 = 1; so a1 + a2 + a3 + a4 = 120 44 = 256; so a ≠ 4 Try a = 3; 3 + 9 + 27 + 81 = 120 a = 3

We have the following set of values: Algebra 1 600 points We have the following set of values: 11, 12, 17,18, 23, 29, and 30 Removing one value causes the mean to decrease by 1.5. Find that value.

29 The mean of the original set of numbers is 20. Algebra 1 600 points Answer The mean of the original set of numbers is 20. The mean of the reduced set of numbers is 20 – 1.5 = 18.5, so the sum of the remaining numbers is 18.5(6) = 111. Since 140 – 111 = 29, the removed number was 29. 29

(the figure is not drawn to scale) Geometry 200 points An angle bisector makes an angle of 71° with the opposite side and intersects another angle bisector at 58°. In degrees, what is the measure of the smallest angle of triangle ABC. (the figure is not drawn to scale) B Holt Geometry Page 442 #20 A C

14° By the triangle sum theorem, x + 58 ° + 71° = 180° and x = 51°. Geometry 200 points Answer By the triangle sum theorem, x + 58 ° + 71° = 180° and x = 51°. Then 51° + 122° + y = 180° and y = 7°. Two angles of the original triangle are now known to measure 2(51°) = 102° and 2(7°) = 14°. The third angle of the triangle measures 180° - 102° - 14° = 64°, so the smallest angle measure is 14°. 14°

Geometry 400 points Each of the small circles in the figure has radius 1 m. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. (Leave the answer in terms of )

2 m2 The large circle has radius 3 m, so its area is (32) = 9 m2. Geometry 400 points Answer The large circle has radius 3 m, so its area is (32) = 9 m2. The seven small circles have a total area of 7( (12)) = 7 m2 So the shaded region has area 9 - 7 = 2 2 m2

Geometry 600 points Two right circular cylinders have the same height but cylinder B has a volume that is 10% less than the volume of cylinder A. What is the ratio of the radius of cylinder B to the radius of cylinder A, in simplest radical form?

𝟎.𝟗 = 𝒓 𝑩 𝟐 𝒓 𝑨 𝟐 take the square root of each side Geometry 600 points Answer 𝑽 𝑩 =𝟎.𝟗𝝅 𝒓 𝑨 𝟐 𝒉 VB = 𝝅 𝒓 𝑩 𝟐 𝒉 𝟎.𝟗𝝅 𝒓 𝑨 𝟐 𝒉 = 𝝅 𝒓 𝑩 𝟐 𝒉 𝟎.𝟗 𝒓 𝑨 𝟐 = 𝒓 𝑩 𝟐 𝟎.𝟗 = 𝒓 𝑩 𝟐 𝒓 𝑨 𝟐 take the square root of each side 𝒓 𝑩 𝒓 𝑨 = 𝟎.𝟗 = 𝟗 𝟏𝟎 = 𝟑 𝟏𝟎 𝟏𝟎 𝟑 𝟏𝟎 𝟏𝟎

Algebra 2 200 points The sum of eight consecutive even integers is 888. Find the smallest of these integers.

Algebra 2 200 points Answer Let the integers be represented by x, x + 2, x + 4, x + 6, x + 8, x + 10, x + 12, and x + 14. The sum of the integers is 8x + 56 = 888 8x = 832 x = 104 104

Algebra 2 400 points A piling is 69 feet long. A portion of the piling is in the ground under the water, part is in the water and the third section is above the water. The water is 12 feet deep. The section of the piling above the water is 3 feet more than 5 times the part that is in the ground under the water. How many feet of piling are above the water?

Let x= number of feet underground. Algebra 2 400 points Answer Let x= number of feet underground. x + 12 + (5x + 3) = 69 6x = 54 x = 9 48 feet

find the exact value of 𝒙 𝟐 + 𝟏 𝒙 𝟐 Algebra 2 600 points If 𝒙+ 𝟏 𝒙 = 𝟐𝟐 , find the exact value of 𝒙 𝟐 + 𝟏 𝒙 𝟐

20 Square both sides of 𝒙+ 𝟏 𝒙 = 𝟐𝟐 to obtain 𝒙+ 𝟏 𝒙 𝟐 = 𝟐𝟐 𝟐 Algebra 2 600 points Answer Square both sides of 𝒙+ 𝟏 𝒙 = 𝟐𝟐 to obtain 𝒙+ 𝟏 𝒙 𝟐 = 𝟐𝟐 𝟐 𝒙 𝟐 +𝟐+ 𝟏 𝒙 𝟐 =𝟐𝟐 𝒙 𝟐 + 𝟏 𝒙 𝟐 =𝟐𝟎 20

Related Technical 200 points A train traveling at a constant rate of 80 km/h enters a tunnel 7 km long at precisely 5:12 a.m. At 5:18 a.m., the caboose exits the tunnel. Find the length of the train. Algebra 2 page 369 Example Pendulum Clock

Related Technical 200 points Answer The time it takes the train to pass through the tunnel is 6 minutes or, 1/10 hour. At 80 km/h, the distance traveled is 80/10 = 8 km. Since the tunnel is 7 km long, the length of the train must be 8 – 7 = 1. 1 km

A garage is creating a sign for their business. Related Technical 400 points A garage is creating a sign for their business. The sign’s dimensions are labeled below. 12” 10” The acrylic material for the sign costs $4.40 / in2. How much will it cost to make the sign, excluding any wasted material?

$7197.39 Atriangle=40 in2 Arectangle=1120 in2 Atrapezoid=380 in2 Related Technical 400 points Answer Atriangle=40 in2 Arectangle=1120 in2 Atrapezoid=380 in2 A1/2 of front tire=39.25 in2 A1/2 of back tire=56.52 in2 Acar=1635.77 in2 Cost=1635.77 X 4.40 $7197.39

Related Technical 600 points A king died, leaving his gold bars to his 6 sons. The youngest son received a certain number of gold bars, the second youngest twice as many, the third youngest three times as many, and so on. The queen ordered each son to give 2 gold bars to every son younger than himself. If, after this, all the sons had the same number of gold bars, how many did each now have? Prentice Hall Geometry Page 562 #18b

14 x = gold bars held by youngest before the trading Related Technical 600 points Answer x = gold bars held by youngest before the trading Thus, before trading, the six sons had x, 2x, 3x, 4x, 5x, and 6x gold bars, respectively, from the youngest to the oldest. After trading, the youngest will have x + 10 bars and the oldest will have 6x – 10 bars. Since each son now has the same number of gold bars, 6x – 10 = x + 10  x = 4 14