Math 70S Integrated Intermediate Algebra

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Sample of what you should know before Math 70S:

1. Given the formula for converting degrees Fahrenheit (F) to degrees Celsius (C), evaluate this formula when $capital f equals negative 13 and capital c equals start fraction 5 over 9 end fraction open parenthesis capital f minus 32 close parenthesis$
- - Solution
    
    $capital c equals start fraction 5 over 9 end fraction open parenthesis negative 13 minus 32 close parenthesis$
    
    $equals start fraction 5 over 9 end fraction open parenthesis negative 45 close parenthesis equals negative 25 degrees Celsius$
2. On a map, 2 cm represents 15 miles. Ryan measures on the map and finds there are 11.5 cm between Springfield and Bloomton. How many miles apart are the two cities?
- - Solution
    
    $x over 11.5 equals 2 over 15$
    
    $15 x equals 2 times 11.5$
    
    $15 x equals 23$
    
    $x equals 23 fifteenths equals 1 and 8 fifteenths miles$
3. A $380 television is on sale for 25% off. If the sales tax is 7.5%, what is the total price you would have to pay for the television?
- - Solution
    
    $380 minus 380 times 0 point 2 5 equals 285$
    
    $285 plus 285 times 0 point 0 7 5 equals 306 point 3 7 5$
    
    $306 dollars and 38 cents$
4. Solve: $3 open parenthesis x minus 2 close parenthesis equals 5 x plus 14$
- - Solution
    
    $3 x minus 6 equals 5 x plus 14$
    
    $negative 2 x minus 6 equals 14$
    
    $negative 2 x equals 20$
    
    $x equals negative 10$
5. Simplify: $24 minus 3 times negative 2 minus 2 squared plus start square root 25 minus 16 end square root$
- - Solution
    
    $24 minus 3 times negative 2 minus 2 squared plus the square root of 9$
    
    $24 minus 3 times negative 2 minus 4 plus 3$
    
    $24 plus 6 minus 4 plus 3$
    
    $30 minus 4 plus 3$
    
    $26 plus 3$
    
    $result= 29$
6. Simplify: $2x open parenthesis 4 x squared minus 6 x plus 2 close parenthesis minus open parenthesis 2 x squared minus 8 x close parenthesis$
- - Solution
    
    $8 x cubed minus 12 x squared plus 4 x minus 2 x squared plus 8 x$
    
    $8 x cubed minus 14 x squared plus 12 x$
Sample of what you will learn in Math 70S:

1. Suppose two salt solutions are to be mixed to create a 25-liter salt solution with 5 gm of salt per liter. The solutions to be mixed contain 2 gm per liter and 10 gm per liter respectively. How much of each solution should be used?
- - Solution
    
    Let x represent the volume in liters of the 2 gm per liter solution.
    
    Let y represent the volume in liters of the 10 gm per liter solution.
    
    Adding the volume in liters, we want $x plus y equals 25$
    
    Adding the mass in grams, we need $2 x plus 10 y equals 5 times 25$
    
    This gives us the system of equations $start set top row x plus y equals 25 bottom row 2 x plus 10 y equals 125 end set$
    
    We can solve $x plus y equals 25$ for y to get $y equals 25 minus x$
    
    Substituting into the second equation, we get $2 x plus 10 open parenthesis 25 minus x close parenthesis equals 125$
    
    $2 x plus 250 minus 10 x equals 125$
    
    $negative 8 x equals negative 125$
    
    $x equals 125 over 8 equals 15 point 6 2 5$
    
    So, $y equals 25 minus start fraction 125 over 8 end fraction equals start fraction 75 over 8 end fraction equals 9 point 3 7 5$
    
    We should use 15.625 liters of the 2 gm per liter solution and 9.375 liters of the 10 gm per liter solution.
2. If you deposit $10,000 into an account that earns a fixed 8% annual interest rate, compounded quarterly, when will the account balance reach $25,000?
- - Solution
    
    Use the compound interest formula $capital a equals capital p open parenthesis 1 plus start fraction r over n end fraction close parenthesis superscript n t baseline$
    
    Here P = 10000, r = 0.08, n = 4, A = 25000, and we're solving for t.
    
    $25000 equals 10000 open parenthesis 1 plus start fraction 0 point 0 8 over 4 end fraction close parenthesis superscript 4 t baseline$
    
    $2 point 5 equals 1 point 0 2 superscript 4 t baseline$
    
    $the natural log of 2 point 5 equals the natural log of 1 point 0 2 to the 4 t$
    
    $the natural log of 2 point 5 equals 4 t times the natural log of 1 point 0 2$
    
    $t equals start fraction the natural log of 2 point 5 over 4 times the natural log of 1 point 0 2 end fraction approximately equal to 11 point 5 6 7 8$
    
    The account balance will reach $25,000 after 12 quarters (or 3 years).
3. The following are root lengths (inches) of garlic cloves after soaking in water for one week. {2.23, 1.71, 1.45, 3.53, 2.57, 1.61, 1.64, 1.67, 1.79, 1.90, 3.04, 2.09, 1.42, 2.16, 1.52, 2.23, 2.26, 1.57, 2.33}. Sketch a boxplot and identify outliers. Describe the distribution of values in terms of center, shape and spread.
- - Solution
    
    First sort the data from smallest to largest:
    
    {1.42, 1.45, 1.52, 1.57, 1.61, 1.64, 1.67, 1.71, 1.79, 1.9, 2.09, 2.16, 2.23, 2.23, 2.26, 2.33, 2.57, 3.04, 3.53}
    
    The number of sample values is n = 19
    
    First quartile: $capital l equals start fraction 25 over 100 end fraction times 19 equals 4 point 7 5 so capital q sub 1 equals 1 point 6 1$
    
    Second quartile: $capital l equals start fraction 50 over 100 end fraction times 19 equals 9 point 5 so capital q sub 2 equals 1 point 9$
    
    Third quartile: $capital l equals start fraction 75 over 100 end fraction times 19 equals 14 point 2 5 so capital q sub 3 equals 2 point 2 6$
    
    $a vertical boxplot with a vertical scale ranging from approximately 1 to 4 a short horizontal line representing the minimum value 1 point 4 2 a vertical line from 1 point 4 2 to 1 point 6 1 a rectangle from 1 point 6 1 to 1 point 9 a rectangle from 1 point 9 to 2 point 2 6 a vertical line from 2 point 2 6 to 3 point 5 3 a short horizontal line maximum value 3 point 5 3$
    
    $I Q R equals capital q sub 3 minus capital q sub 1 equals 2 point 2 6 minus 1 point 6 1 equals 0 point 6 5$
    
    $1 point 5 times I Q R equals 1 point 5 times 0 point 6 5 equals 0 point 9 7 5$
    
    $capital q sub 1 minus 0 point 9 7 5 equals 1 point 6 1 minus 0 point 9 7 5 equals 0 point 6 3 5$
    
    There are no sample values less than 0.635.
    
    $capital q sub 3 plus 0 point 9 7 5 equals 2 point 2 6 plus 0 point 9 7 5 equals 3 point 2 3 5$
    
    3.53 is more than 3.235, so 3.53 is an outlier.
    
    The center is 1.9. The boxplot has a center of 1.9 and is skewed to the right.

Math 70S Integrated Intermediate Algebra

Sample of what you should know before Math 70S:

Sample of what you will learn in Math 70S: