Question 1

Carole's car averages 17.4 mi/gal in city driving and 25.7 mi/gal in highway driving. If she drove a total of 523.9 mi on 22 gal of gas, how many of the gallons were used for city driving?

Accepted Answer

The answer of Carole's car averages 17.4 mi/gal in city...

Question 2

A company makes 3 types of cable. Cable A requires 3 black, 3 white, and 2 red wires. B requires 1 black, 2 white, and 1 red. C requires 2 black, 1 white, and 2 red. The company used 95 black, 100 white and 80 red wires.
How many of each type of cable were made?

Accepted Answer

Let the number of cables A, B, and C made be a, b, and c respectively. We have the following system of equations based on the problem statement:- For black wires: 3a + 1b + 2c = 95- For white wires: 3a + 2b + 1c = 100- For red wires: 2a + 1b + 2c = 80Solving this system of equations, we find that a = 15, b = 20, and c = 15, which corresponds to option B: 15 A; 20 B; 15 C.

Question 3

The following table shows the number of dog collars, in thousands, produced by Allied Pet Products in the three years considered. Find a function of the form $f ( x ) = a x ^ { 2 } + b x + c$ that fits this data. Using this function, what would you predict the dog collar production of Allied Pet Products to be in year 27 ?

Accepted Answer

The answer of The following table shows the number of...

Question 4

A company makes 3 types of cable. Cable A requires 3 black, 3 white, and 2 red wires. B requires 1 black, 2 white, and 1 red. C requires 2 black, 1 white, and 2 red. The company used 100 black, 110 white and 90 red
Wires. How many of each type of cable were made?

Accepted Answer

To solve this, set up equations based on the information given:For black wires: 3A + 1B + 2C = 100For white wires: 3A + 2B + 1C = 110For red wires: 2A + 1B + 2C = 90Solving these equations simultaneously gives A = 10, B = 30, C = 20.

Question 5

A $74,000 trust is to be invested in bonds paying 8%, CDs paying 5%, and mortgages paying 9%. The bond and CD investment together must equal the mortgage investment. To earn a $5930 annual income from the
Investments, how much should the bank invest in bonds?

Accepted Answer

The total investment is $74,000, and the annual income goal is $5,930. Let's denote the amount invested in bonds as B, in CDs as C, and in mortgages as M. We have two conditions: B + C = M, and the total investment is B + C + M = $74,000. The income from these investments is 0.08B + 0.05C + 0.09M = $5,930. Given B + C = M, we can substitute M with B + C in the total investment equation: B + C + (B + C) = $74,000, simplifying to 2B + 2C = $74,000, or B + C = $37,000. Using the income equation, we substitute M with B + C, getting 0.08B + 0.05C + 0.09(B + C) = $5,930. Since B + C = $37,000, we can replace C with $37,000 - B in the income equation, leading to a single variable equation in terms of B. Solving this equation directly for B (given the constraints and the need for a specific solution) without the specific algebraic steps here (due to the format constraint), we find that B = $25,000 meets the conditions for the investment strategy to achieve the desired annual income.

Question 6

Linda invests $25,000 for one year. Part is invested at 5%, another part at 6%, and the rest at 8%. The total income from all 3 investments is $1600. The combined income from the 5% and 6% investments is the same as
The income from the 8% investment. Find the amount invested at each rate.

Accepted Answer

The income from the 5% and 6% investments is equal to the income from the 8% investment. Solving the equations: 0.05x + 0.06y + 0.08z = 1600 and 0.05x + 0.06y = 0.08z with x + y + z = 25000, we find the amounts are $10,000 at 5%, $5000 at 6%, and $10,000 at 8%.

Question 7

The perimeter of a triangle is 70 cm. The triangle is isosceles now, but if its base were lengthened by 7 cm and each leg were shortened by 4 cm, it would be equilateral. Find the length of the base of the original triangle.

Accepted Answer

The original triangle has two equal sides (legs) and a base. Let the base be x cm and each leg be y cm. The perimeter equation is x + 2y = 70. If the base is increased by 7 cm and each leg is decreased by 4 cm, the triangle becomes equilateral, so x + 7 = y - 4. Solving these equations gives x = 16 cm.

Question 8

The sum of a student's three scores is 223. If the first is 11 points more than the second, and the sum of the first two is 43 more than twice the third, what was the first score?

Accepted Answer

Let the scores be x, y, and z. We have x + y + z = 223, x = y + 11, and x + y = 2z + 43. Solving these equations, we find x = 87.

Question 9

The perimeter of a rectangle is 44 cm. The length is 12 cm longer than the width. What are the length and width of the rectangle?

Accepted Answer

The answer of The perimeter of a rectangle is 44...

Question 10

The following table shows the number of dog collars, in thousands, produced by Allied Pet Products in the three years considered. For a function of the form $f ( x ) = a x ^ { 2 } + b x + c$ that fits this data, what is the value of  b  ?
$\begin{array} { c | c } 
\text { Year } & \begin{array} { c } 
\text { Number of dog collars } \
\text { (in thousands) }
\end{array} \
\hline 0 & 20 \
\hline 6 & 14 \
\hline 18 & 23.6
\end{array}$

Accepted Answer

The answer of The following table shows the number of...

Question 11

Two different gasohol mixtures are available. One contains 5% alcohol and the other 12% alcohol. How much of each should be mixed to obtain 1500 gal of gasohol containing 10% alcohol?

Accepted Answer

To solve this, we use the equation for mixture problems: amount of pure substance = concentration * total amount. Let x be the amount of 5% alcohol mixture, then (1500 - x) is the amount of 12% alcohol mixture. The equation is 0.05x + 0.12(1500 - x) = 0.10 * 1500. Solving for x gives x = 429 gal of 5% and 1500 - 429 = 1071 gal of 12%.

Question 12

A grain dealer sold to one customer 2 bushels of wheat, 9 of corn, and 6 of rye for $45.20; to another, 9 of wheat, 6 of corn, and 2 of rye for $35.60; and to a third, 6 of wheat, 2 of corn, and 9 of rye for $45.00. What was the price
Per bushel for corn?

Accepted Answer

The problem can be solved by setting up a system of equations based on the information given and solving for the price per bushel of corn. Let w, c, and r represent the prices per bushel of wheat, corn, and rye, respectively. The equations based on the transactions are:1. 2w + 9c + 6r = 45.202. 9w + 6c + 2r = 35.603. 6w + 2c + 9r = 45.00Solving this system of equations, we find that c (the price per bushel of corn) is $2.40.

Question 13

Bob fenced in a rectangular garden in his yard. The length of the rectangle is 5 ft longer than the width and the perimeter is 66 ft. What is the width of the rectangle?

Accepted Answer

The width of the rectangle can be found by setting up the equation for the perimeter of a rectangle, P = 2l + 2w, where l is the length and w is the width. Given that the length is 5 ft longer than the width (l = w + 5) and the perimeter is 66 ft, we substitute and solve for w: 66 = 2(w + 5) + 2w, simplifying to 66 = 4w + 10, then to 56 = 4w, and finally w = 14 ft.

Question 14

The following table shows the per capita cumquat consumption, in pounds, in the Virgin Islands in three different years. Find a function of the form $f ( x ) = a x ^ { 2 } + b x + c$ that fits this data. Using this function, what would you predict the per capita cumquat consumption to be in year 14 ?

Accepted Answer

The answer of The following table shows the per capita...

Question 15

Michael's bank contains only nickels, dimes, and quarters. There are 67 coins in all, valued at $5.25. The number of nickels is 1 short of being three times the sum of the number of dimes and quarters. How many dimes are in
The bank?

Accepted Answer

Let n, d, and q represent the number of nickels, dimes, and quarters, respectively. We have three equations: n + d + q = 67, 0.05n + 0.10d + 0.25q = 5.25, and n = 3(d + q) - 1. Solving these equations, we find that there are 10 dimes.

Question 16

A contractor mixes concrete from bags of pre-mix for small jobs. How many bags with 6% cement should he mix with 3 bags of 12% cement to produce a mix containing 7% cement?

Accepted Answer

The answer of A contractor mixes concrete from bags of...

Question 17

The perimeter of a rectangle is 64 m. If the width were doubled and the length were increased by 5 m, the perimeter would be 100 m. What are the length and width of the rectangle?

Accepted Answer

The original perimeter equation is 2(l + w) = 64, simplifying to l + w = 32. The modified perimeter equation is 2(l + 5 + 2w) = 100, simplifying to l + 2w = 45. Solving these equations gives l = 19 and w = 13.

Question 18

Twice the water flow in the hot-water pipe is the same as three times the flow in the cold-water pipe. The combined flow is  1500 L/hr. What is the flow in each pipe?

Accepted Answer

The answer of Twice the water flow in the hot-water...

Question 19

Best Rentals charges a daily fee plus a mileage fee for renting its cars. Barney was charged $90 for 3 days and 300 miles, while Mary was charged $163 for 5 days and 600 miles. What does Best Rental charge per day and
Per mile?

Accepted Answer

Barney was charged $90 for 3 days and 300 miles, so the daily fee is $90/3 = $30. The mileage fee is ($90 - $30)/300 = $0.20 per mile.Mary was charged $163 for 5 days and 600 miles, so the daily fee is $163/5 = $32.60. The mileage fee is ($163 - $32.60)/600 = $0.2183 per mile.So the correct answer is A) $17 per day; 13¢ per mile.

Question 20

The following table shows the number of dog collars, in thousands, produced by Allied Pet Products in the three years considered. For a function of the form $f ( x ) = a x ^ { 2 } + b x + c$ that fits this data, what is the value of c?
$\begin{array} { c | c } 
\text { Year } & \begin{array} { c } 
\text { Number of dog collars } \
\text { (in thousands) }
\end{array} \
\hline 0 & 20 \
\hline 6 & 14 \
\hline 18 & 23.6
\end{array}$

Accepted Answer

The value of $c$ in the function $f(x) = ax^2 + bx + c$ represents the value of the function when $x = 0$. From the table, when the year is 0, the number of dog collars produced is 20,000. Therefore, $c = 20$.

Quiz 6: Systems and Matrices

Carole's car averages 17.4 mi/gal in city driving and 25.7 mi/gal in highway driving. If she drove a total of 523.9 mi on 22 gal of gas, how many of the gallons were used for city driving?

A company makes 3 types of cable. Cable A requires 3 black, 3 white, and 2 red wires. B requires 1 black, 2 white, and 1 red. C requires 2 black, 1 white, and 2 red. The company used 95 black, 100 white and 80 red wires. How many of each type of cable were made?

A company makes 3 types of cable. Cable A requires 3 black, 3 white, and 2 red wires. B requires 1 black, 2 white, and 1 red. C requires 2 black, 1 white, and 2 red. The company used 100 black, 110 white and 90 red Wires. How many of each type of cable were made?

A $74,000 trust is to be invested in bonds paying 8%, CDs paying 5%, and mortgages paying 9%. The bond and CD investment together must equal the mortgage investment. To earn a $5930 annual income from the Investments, how much should the bank invest in bonds?

Linda invests $25,000 for one year. Part is invested at 5%, another part at 6%, and the rest at 8%. The total income from all 3 investments is $1600. The combined income from the 5% and 6% investments is the same as The income from the 8% investment. Find the amount invested at each rate.

The perimeter of a triangle is 70 cm. The triangle is isosceles now, but if its base were lengthened by 7 cm and each leg were shortened by 4 cm, it would be equilateral. Find the length of the base of the original triangle.

The sum of a student's three scores is 223. If the first is 11 points more than the second, and the sum of the first two is 43 more than twice the third, what was the first score?

The perimeter of a rectangle is 44 cm. The length is 12 cm longer than the width. What are the length and width of the rectangle?

Two different gasohol mixtures are available. One contains 5% alcohol and the other 12% alcohol. How much of each should be mixed to obtain 1500 gal of gasohol containing 10% alcohol?

A grain dealer sold to one customer 2 bushels of wheat, 9 of corn, and 6 of rye for $45.20; to another, 9 of wheat, 6 of corn, and 2 of rye for $35.60; and to a third, 6 of wheat, 2 of corn, and 9 of rye for $45.00. What was the price Per bushel for corn?

Bob fenced in a rectangular garden in his yard. The length of the rectangle is 5 ft longer than the width and the perimeter is 66 ft. What is the width of the rectangle?

Michael's bank contains only nickels, dimes, and quarters. There are 67 coins in all, valued at $5.25. The number of nickels is 1 short of being three times the sum of the number of dimes and quarters. How many dimes are in The bank?

A contractor mixes concrete from bags of pre-mix for small jobs. How many bags with 6% cement should he mix with 3 bags of 12% cement to produce a mix containing 7% cement?

The perimeter of a rectangle is 64 m. If the width were doubled and the length were increased by 5 m, the perimeter would be 100 m. What are the length and width of the rectangle?

Twice the water flow in the hot-water pipe is the same as three times the flow in the cold-water pipe. The combined flow is 1500 L/hr. What is the flow in each pipe?

Best Rentals charges a daily fee plus a mileage fee for renting its cars. Barney was charged $90 for 3 days and 300 miles, while Mary was charged $163 for 5 days and 600 miles. What does Best Rental charge per day and Per mile?