Math 005A Prerequisite Material Answer Key

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1 Math 005A Prerequisite Material Answer Key 1. a) P = 4s (definition of perimeter and square) b) P = l + w (definition of perimeter and rectangle) c) P = a + b + c (definition of perimeter and triangle) d) C = πr (definition of circumference) e) s = rθ (definition of arc length, radius, and central angle). a) A = s (definition of area of a square) b) A = lw (definition of area of a rectangle) c) A = πr (definition of area of a circle) d) A = 1 r θ (definition of area of a sector, radius, and central angle) e) A = 1 bh, A = 1 absinc, A = s ( s a )( s b) ( s c), where s = a + b + c (standard area of a triangle formula, SAS area formula, Heron s Formula) f) A = h B + b (Area of a trapezoid formula) g) S = 4πr (Surface area of sphere formula) 3. a) V = lwh (definition of volume of rectangular solid) b) V = 1 bhl (Volume of right prism with triangular base formula) c) V = lh B + b (Volume of right prism with trapezoidal base formula) d) V = πr h (Volume of a cylinder formula) e) V = 1 3 πr h (Volume of a cone formula) f) V = 1 3 lwh (Volume of a pyramid formula) g) V = 4 3 πr3 (Volume of a sphere formula) (note: 3 a, b, c, d are all examples of right prisms, which have the general formula Volume = Area of Base Height of the prism. In 3b and c the height of the prism is denoted with l and the h denotes the height of the base) 4. a) x 3 + 4x 4x 9 (distributive property, combining like terms) b) 8x 3 4x 48x + 4 (distributive property, combining like terms)
2 c) 4x 5 3x + 4 d) (distributive property, combining like terms) 4x +10x 4 x + ( x ) ( x + 3) (factoring, finding a common denominator, equivalent fractions, distributive property, combining like terms) e) 4x 1x + 9 (FOIL or distributive property, combining like terms) f) tan x sec x (Pythagorean trig identity, quotient trig identity, reciprocal trig identity) g) sin x (double angle formula for sine) h) x (factoring) 5. a) y = 1 5 x (slope of a line, point slope formula for the equation of a line) b) y = 1 x 3 (slopes of perpendicular lines, point slope formula for the equation of a line, isolating a variable) c) y = x + 4 (slopes of parallel lines, point slope formula for the equation of a line) 6. a) 9 (distance formula) b) d(x) = x 5x 10 x + 34 (distance formula, substitution you are substituting square root x for y in the distance formula, FOIL/distributive property, combining like terms) 7. For a right triangle with legs of length a and b and hypotenuse c, a + b = c. (Pythagorean theorem, definition of right triangle, definition of hypotenuse) 8. a) d(t) = 60t (distance = rate x time) b) d(t) = 0t 13 (modeling with functions by placing the initial starting point at the origin and expressing the position of each car as a point with two coordinates with t as a variable, distance formula, simplifying polynomials, common factor, simplifying expressions involving square root) 9. a) A(x) = P 16 simplifying) (isolating for a variable, area of a square, substitution, b) h = 8 5 r (similar triangles, proportions, solving for a variable)
3 c) h = 6 5 b (similarity, proportions, solving for a variable) 10. a) x 5 (Multiplication law of exponents) b) x 6 (Power of a power law of exponents) c) x (Division law of exponents) d) is already simplified e) x (negative exponents, multiplying a fraction by a constant) f) x (Division law of exponents) 11. a) x (Fractional exponents) b) x (Fractional exponents, index of a root) c) 1 x (Fractional exponents, negative exponents) d) x (Fractional exponents, negative exponents, multiplying a fraction by a constant) 1. a) x 1/ (Fractional exponents) b) x 1/4 (Fractional exponents, negative exponents) c) x 3/ (Fractional exponents, division law of exponents, fraction subtraction) d) x +1 x 1/ (Fractional exponents, finding a common denominator, Multiplication law of exponents) 13. a) 3( 1+ 3x) ( 1 3x) (GCF, difference of two squares) b) x( x +10) ( x ) (GCF, factoring quadratics) (grouping, sum of two cubes) c) ( x + 5) 4x 10x + 5 x +1 d) x 3 +1 (grouping, GCF) e) ( x + ) ( 5x + 9) (GCF, distributive law, combining like terms) f) ( 3x 5) ( 5x + 9) (Difference of two cubes using substitution, FOIL/distributive law, combining like terms) g) 6( x + 5) 3 ( x 1) ( x +1) (GCF, distributive law, combining like terms, GCF)
4 14. a) 3x 8x + 3 x +1 or b) x 5x 9 x + 9 c) 4 3x 1 x +1 ( 3x 1) ( x + 3) ( x +1) (distributive law, combining like terms) (distributive law, combining like terms, factoring GCF) 3 (factoring GCF, distributive law, combining like terms) d) x + 3/ (LCD, multiplying roots with the same base, distributive law, 1+ x combining like terms, converting radicals to exponential form, multiplication law of exponents) e) x 4 x + 4 3/ (either LCD or factoring trick like terms, laws of exponents) f) 1 ( x +1 ) 1/ ( 5x + ) g) 4x /3 ( x 1) (factoring GCF) (factoring GCF, combining like terms) a 1 + a 1 = a 1 [a +1], combining h) 1 x/3 1 x5/3 (distributive law, laws of exponents) 15. a) No solution (rational expression = 0 when numerator is 0, common denominator, multiplication law of exponents, distributive law, combining like terms, solving a linear equation, check answers, cannot have negative under square root) b) x = 1 3 (rational expression = 0 when numerator is 0, common denominator, multiplication law of exponents, combining like terms, solving a quadratic, check answers, cannot have negative under square root) c) x = 5, 1, 1 (factor out GCF, distributive law, combining like terms, solving linear equations)
5 d) x = 5, 1 (common denominator for RHS, multiply both sides by LCM of denominators, distributive law, combining like terms, factoring, solving linear equations) e) x = 1 (move everything to one side, factor by grouping, solving linear equation) f) x = π 6 + kπ, π + kπ, 5π 6 + kπ, where k is an integer (double angle formula for sine, factor, solving a trig equation, periodicity of sine and cosine) g) x = π ( 1+ k)π + kπ,, 5π + kπ, where k is an integer (double angle 6 6 formula for cosine (the one that has sine), distributive law, factoring with substitution, solving trig equations, periodicity of sine and cosine) / h) x = kπ, + kπ, where k is an integer (solve for the argument and then 0 solve for the variable) i) x = k + 1 π, where k is an integer (solve for the argument and then solve for the variable) j) x = kπ, 0/, 5/ + kπ, where k is an integer (Pythagorean identity, then 4 4 solve by factoring) a) x + h + x (multiplying by 1 (i.e. top and bottom by the conjugate), FOIL or difference of two squares formula, combining like terms) 1 b) x x + h x + x + h (finding a common denominator, multiplying top and bottom by the conjugate, difference of two squares formula, distributive law, combining like terms) 17. a) xy (negative exponents, finding a common denominator, factoring) x b) ( x +1) (LCM, multiplying by 1, distributive law, combining like terms, factoring)
6 5x 4 c) ( x + 3) x +1 ( x ) (LCM, distributive law, dividing x+3 is the same as multiplying by the reciprocal 1/(x+3)) 18. a) 7, ( 3, 7) (substitution, using a function to find the y coordinate of a point on the graph of a function) b) 5/3 3 or 4, 4, 5/3 (substitution, using a function to find the y coordinate of a point on the graph of a function) 19. a) 4x h + 3 (substitution, distributive law, combining like terms, factoring) x + h + 4 b) (substitution, distributive law, combining like terms, x + ( x + h + ) factoring) 1 c) (substitution, multiplying top and bottom by conjugate, x + h 3 + x 3 distributive law, combining like terms) 0. Use desmos.com or wolframalpha.com # x if x < 0 1. a) f (x) = $ % x if x 0 (definition of absolute value) # x 5 if x < 5 % b) f (x) = $ % x + 5 if x 5 &% (solving linear inequalities involving absolute value) # cosx sin x if 0 x < π % c) (see #15g) h(x) = 6 $ % sin x cosx if π 6 x π &% (solving a trigonometric equation, double angle formula for cosine, factoring with substitution) $ & d) y = x 4 if x or x % '& 4 x if < x < (factoring, solving quadratic inequalities involving absolute value)
7 . a) { x x ±} (factoring, can t have 0 in denominator) b) all real numbers (understanding an irreducible polynomial has no real roots, can t have 0 in denominator) { } c) x x 3 the square root) d) x x > 3 { } (solving a linear inequality, can t have negative numbers under (solving a linear inequality, can t have negative numbers under the square root, can t have 0 in denominator) e) { x x > 4} (domain of ln, solving a linear inequality) f) { x x < 1 or x > 6} (factoring, solving a quadratic inequality, can t have negative numbers under the square root) k +1 g) x x pi, where k is an integer (domain of tangent function, periodicity of tangent, being able to come up with an algebraic description given a pattern) h) x x 3 { } (denominator can t be 0) i) all real numbers (domain of a linear function) 3. a) y = 5 x (understanding the difference between the positive and negative square root) b) y = x + 3 (understanding the difference between the positive and negative square root) 4. a) sin θ + cos θ =1, tan θ +1= sec θ, 1+ cot θ = csc θ b) sinx = sin x cos x (double angle formula for sine) c) cosx = cos x sin x cos x 1 1 sin x (double angle formula for cosine) d) c = a + b abcosc e) sin0 = 0 f) sin( α + β) = sinα cosβ + cosα sin β g) cos( α + β) = cosα cosβ sinα sin β h) see next page (sum formula for sine) (sum formula for cosine)
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