ALGEBRA 1GEOMETRYALGEBRA 2RELATED

TECHNICAL

2015 Vocational Math Meet: Round 2

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Algebra 1200 points

Given the following system of equations

9x – 6y = 216x – 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 = 73x – 2y = 0.5k

0.5k =7k = 14

Algebra 1 100 pointsAlgebra 1400 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

Algebra 1 400 points Answer

a0 = 1; so a1 + a2 + a3 + a4 = 120

44 = 256; so a ≠ 4Try a = 3; 3 + 9 + 27 + 81 = 120

a = 3

Algebra 1600 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.

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

Geometry200 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

A C

Geometry200 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°

Geometry400 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 )

Geometry400 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

Geometry600 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?

Geometry600 points Answer

VB = = =

= take the square root of each side

𝟑√𝟏𝟎𝟏𝟎

Algebra 2200 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 2400 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?

Algebra 2 400 points Answer

Let x= number of feet underground.

x + 12 + (5x + 3) = 69

6x = 54

x = 9

48 feet

Algebra 2600 points

If ,

find the exact value of

Algebra 2 600 points Answer

Square both sides of to obtain

20

Related Technical200 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.

Related Technical200 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

Related Technical400 points

A garage is creating a sign for their business.The sign’s dimensions are labeled below.

The acrylic material for the sign costs $4.40 / in2. How much will it cost to make the sign, excluding any wasted material?

12” 10”

Related Technical400 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 Technical600 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?

Related Technical600 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