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Algebra 1 Geometry Algebra 2Related Technical

Jeopardy Round 2

Algebra 1

100 points

When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number?

Algebra 1 100 points

Answer

Moving the decimal point to the right four places is equivalent to multiplying the number by 10,000, that is, 104. Then if x is the original number, we know that 104x = 4(1/x). Therefore, x2 = 4/10000, x=0.02


The difference between the squares of two consecutive odd integers is 128. What is the product of the two integers?


Answer

Let x and x + 2 be the two consecutive odd integers. Then 128 = (x + 2)2 – x2

128 = 4x + 44x = 124 x = 31 and x + 2 = 33.

The product of the two integers is 31(33)= 1023.


A series of 7 books was published at 9-year intervals. When the 7th book was published, the sum of the publication years was 13601. In what year was the 4th book published?


Answer

The year of publication of the 4th book is the average of the publication years of all 7 books. This average is 13601 / 7 = 1943


If a + b = 6 a

and b + c = 9 c

Compute the numerical value of a/c.


Answer

Since 6 = a + b = 1 + b a a

and since 9 = b+ c = b + 1 c c

we have b/a = 5 and b/c = 8 ba = cc b

a = 8/5

Geometry

100 points

In the figure below, ABCD is a rectangle and DE = DC. Given AD = 5 and BE = 3, compute DE.

Geometry 100 points

Answer

We label the figure with the given lengths, let x be the value common to DE and DC, and let y = AE. Then x = y + 3, since ABCD is a rectangle; and y2 + 25 = x2, by the Pythagorean theorem. Substituting x – 3 for y in the second equation gives us X2 = (x – 3)2 + 25 = x2 –6x + 34so 6x = 34, and therefore x = 17/3 units

Geometry 200 points

The area of the square below is 64cm2. If C is the midpoint of AD and if B is the midpoint of AC, what is the area of the shaded triangle?

Geometry 200 points

Answer

Since the square’s area is 64cm2, each of its sides has length 8cm, so BC = 2cm. BC is the base of the triangle, and its height is 8cm, so the area of the

triangle is (1/2)(2)(8), or 8cm2.

Geometry 300 points

When the length of each of the edges of a cube is increased by 1 centimeter, the cube’s total surface area increases by 78 square centimeters. What is the length of an edge of the original cube?

Geometry 300 points

Answer

Let x be the length of each edge of the original cube in centimeters. Then the surface area of the original cube is 6x2 square centimeters and the surface area of the larger cube is 6(x + 1)2 square centimeters, so 78 + 6x2 = 6(x + 1)2.

78 = 6(x + 1)2 – 6x2

= 12x + 6 x = 72/12 x = 6

centimeters

Geometry 400 points

Given that ABCD is a square, AF = BG = 5, and BF = CH = DE = 12, compute the area of EFGH.

Geometry 400 points

Answer

Referring to the diagram, the area of ABCD is 289 square units, so the area of EFGH is 289 minus the sum of the areas of triangles FAE, FBG, GCH and EDH. The areas of these triangles are 25/2, 30,72, and 30 respectively, so the area of EFGH is 289 – ( 132 + 25/2) = 157 – 25/2 = (314 – 25) 2=289 2= 144.5 square units

Algebra 2

100 points

What are all values of x which satisfy

4x3

+ 5x2

– 6x = 1


Answer

Since the right-hand side is 1, the exponent on the left-hand side must be 0.

Thus, 0 = x3 + 5x2 – 6x = (x)(x – 1)(x + 6), and

X = 0, 1, -6


A survey of 50 students found that 30 had cats, 25 had dogs, 5 had white mice, 16 had both dogs and cats, 4 had both dogs and mice, 2 had both cats and mice, and only 1 had all three kinds of pets. How many students had no pets of these types?


Answer

We can use a Venn diagram to keep track of the data, where D is the set of students that had dogs, C is the set of students with cats, and M is the set of students with mice. We must be sure that the number of students in D is 25, the number in C is 30, the number in D∩C is 16, the number in M∩C is 2, the number in M∩D is 4, and the number in D∩M∩C is 1. Carefully determining the number of students in each component of the diagram, we see that only thirty-nine students have dogs, cats, mice or some combination of more than one of these, so eleven students have none of these pets.


I f ((5/4)4) (5/4)5 = ((5/4)5)x,

What is the real value of x?


Answer


Bouquets are priced on the basis of the numbers and types of flowers used. What is the price of the fourth bouquet?


Answer

Let x, y, and z denote the respective prices of flowers. From the given information, we have three equations:2x + y + z = 4.20x + 2y + z = 3.802x + 2z = 4.80The solution is x = 1.10 y = 0.70 and z = 1.30The fourth bouquet is priced at 2x + 3y +z = 2(1.10) +3(0.70) + 1.30

= $5.60

Related Technical

100 points

What percent of the lot illustrated is taken up by the house and driveway?

Related Technical 100 points

Answer

(2150 + 820)(90)(150)

= 2970 13500

= 0.22(100)

= 22%


If water runs into a 25ft diameter cylindrical tank at a rate of 20 gallons/minute, how long (hours and nearest minute) will it take to fill the tank to a depth of 4’ – 6” ? (7.5gal/cu.ft)


Answer

V = (π)(12.5)2(4.5) = 2207.81ft3 (7.5) = 16558.59 gal = 16558.59/ 20 gal/min = 828 min/60 = 13hrs 48minutes


The turning radius of a car is 19 feet. How many feet further does one rear wheel travel than the other in making a 90-degree turn? The standard track or tread equals 56 inches, or 4 2/3 feet. (answer rounded to nearest tenth)


Answer

Wheel 2 (2(16.66)(3.14))/4= 26.17

Wheel 1 (2(21.33)(3.14)/4= 33.49

Wheel 1 – Wheel 2 = 7.3ft


A mechanic’s commission is increased from 45% to 48% at a flat-rate charge of $27.50 an hour. How much more can he expect in his pay if he turned in 49 hours the first week?


Answer

.45 x 27.50 x 49 = 606.375.48 x 27.5 x 49 = 646.8646.8 – 606.375 = $40.43

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