Mathematical Modeling - ACT Math

1. What is the midpoint formula between points $ ($x_1$,$y_1$) $ and $ ($x_2$,$y_2$) $?

   $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average of x-coordinates and y-coordinates.

2. What is the formula for the surface area of a sphere?

   Surface Area = $4\pi r^2$. Four times the area of a great circle.

3. What is the weighted average model for mean of values $x_1,x_2$ with weights $w_1,w_2$?

   $\frac{w_1x_1+w_2x_2}{w_1+w_2}$. Sum of weighted values divided by total weights.

4. What is the slope formula between points $(x_1,y_1)$ and $(x_2,y_2)$?

   $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run between two points.

5. What is the formula for the area of a trapezoid?

   Area = $\frac{1}{2}$(base₁ + base₂) × height. Average of parallel sides times height.

6. Find the slope of the line through points (2, 3) and (5, 11).

   Slope = $\frac{8}{3}$. Change in y over change in x: $\frac{11-3}{5-2} = \frac{8}{3}$.

7. Given $A=50(1.1)^t$, what is $A$ when $t=0$?

   $50$. Initial value when exponent equals zero.

8. What is the formula for the sum of the interior angles of a polygon?

   Sum = $(n-2) × 180$. Each interior angle contributes to $(n-2) × 180$ degrees total.

9. Find the area of a circle with radius 4.

   Area = $16\text{π}$. Use $A = \text{π}r^2$ with $r = 4$.

10. Identify the formula for the circumference of a circle.

    Circumference = $2\pi r$. Perimeter formula where $r$ is the radius.

11. Identify the best model type when $y$ changes by a constant percent per unit increase in $x$.

    Exponential model. Constant percent change indicates exponential relationship.

12. If $y$ varies inversely with $x$ and $y=5$ when $x=8$, what is $y$ when $x=10$?

    $y=4$. Since $k=xy=40$, when $x=10$, $y=4$.

13. A quantity decays $20\%$ per step; write the factor $b$ in $y=a b^x$.

    $b=0.8$. Decrease of 20% means 80% remains.

14. What is the exponential growth/decay model with initial value $a$ and factor $b$?

    $y=ab^x$. Where $a$ is initial value and $b$ is growth/decay factor.

15. What is the quadratic formula?

    $x = \frac{-b \, ± \, \text{√}(b^2 - 4ac)}{2a}$. Solves $ax^2 + bx + c = 0$ for any quadratic.

16. What is the formula for the area of a rectangle?

    $Area = length \times width$. Base formula for calculating space inside a rectangle.

17. Calculate the Fahrenheit for 25°C.

    $F = 77$. Apply conversion: $\frac{9}{5}(25) + 32 = 77$.

18. Identify the best model type when the change in $y$ is a constant amount per unit increase in $x$.

    Linear model. Constant rate of change indicates linear relationship.

19. What is the point-slope form of a line with slope $m$ through $(x_1,y_1)$?

    $y-y_1=m(x-x_1)$. Uses known point and slope to define the line.

20. Convert 120 degrees to radians.

    $\frac{2\pi}{3}$ radians. Multiply by $\frac{\pi}{180}$: $120 \times \frac{\pi}{180} = \frac{2\pi}{3}$

21. How do you find the volume of a rectangular prism?

    Volume = length × width × height. Multiply all three dimensions for 3D space.

22. State the formula for the Pythagorean theorem.

    $a^2 + b^2 = c^2$. Relates the sides of a right triangle.

23. What is the meaning of the intercept $b$ in a linear model $y=mx+b$?

    Value of $y$ when $x=0$. Where the line crosses the y-axis.

24. What is the area of a triangle with base 6 and height 4?

    Area = 12. Triangle area: $\frac{1}{2}(6)(4) = 12$.

25. Find the equation of a line with slope 2 through point (1, 3).

    $y - 3 = 2(x - 1)$. Point-slope form: $y - y_1 = m(x - x_1)$.

26. Find the area of a rectangle with length 7 and width 3.

    Area = 21. Rectangle area: $7 \times 3 = 21$.

27. What is the vertex form of a quadratic model with vertex $(h,k)$?

    $y=a(x-h)^2+k$. Parabola with vertex at $(h,k)$ and stretch factor $a$.

28. Identify the best model type when $xy$ is constant for varying $x$ and $y$.

    Inverse variation. Product remains constant as variables change.

29. What is the formula for the area of a sector of a circle?

    Area = $\frac{\theta}{360} \times \pi r^2$. Fraction of full circle times total area.

30. Determine the y-intercept of the equation $y = 3x + 4$.

    $4$. The constant term in $y = mx + b$ form.

31. What is the compound interest model for principal $P$ at rate $r$ compounded $n$ times for $t$ years?

    $A=P\left(1+\frac{r}{n}\right)^{nt}$. Accounts for compounding frequency and time.

32. Solve for $y$: $3y - 2 = 10$.

    $y = 4$. Add 2 to both sides, then divide by 3.

33. What is the average rate of change of $f$ from $x=a$ to $x=b$?

    $\frac{f(b)-f(a)}{b-a}$. Slope of secant line between two points.

34. Find the linear model $y=mx+b$ through $(0,3)$ and $(4,11)$.

    $y=2x+3$. Slope $m=\frac{11-3}{4-0}=2$, y-intercept $b=3$.

35. Calculate the volume of a sphere with radius 2.

    Volume = $\frac{32}{3}\text{π}$. Sphere volume: $\frac{4}{3}\text{π}(2^3) = \frac{32}{3}\text{π}$.

36. For the exponential model $y=200(1.05)^t$, what is the growth factor per unit $t$?

    $1.05$. The base of the exponential function.

37. Identify the formula for the surface area of a cylinder.

    Surface Area = $2πr(h + r)$. Two circular ends plus curved side surface.

38. State the quadratic formula.

    $x = \frac{-b \, \text{±} \, \text{√}(b^2 - 4ac)}{2a}$. Solves $ax^2 + bx + c = 0$ for any quadratic.

39. Identify the formula for the slope of a line given two points.

    Slope = $\frac{y_2 - y_1}{x_2 - x_1}$. Rise over run between two coordinate points.

40. What is the formula for the perimeter of a rectangle?

    Perimeter = 2(length + width). Sum of all four sides of a rectangle.

41. What is the formula for the volume of a sphere?

    Volume = $\frac{4}{3}πr³$. Four-thirds times pi times radius cubed.

42. What is the formula for the volume of a cone?

    Volume = $\frac{1}{3} \pi r^2 h$. One-third of cylinder volume formula.

43. What is the formula for the volume of a cylinder?

    Volume = $\text{πr}^2 \times \text{height}$. Base area times height for cylinder volume.

44. What is the formula for the future value of an annuity?

    $FV = Pmt \times \frac{(1 + r)^n - 1}{r}$. Sum of equal payments growing at rate $r$ for $n$ periods.

45. What is the standard form of a quadratic model in $x$?

    $y=ax^2+bx+c$. General form of quadratic with coefficients $a$, $b$, $c$.

46. What is the formula for compound interest?

    $A = P(1 + \frac{r}{n})^{nt}$. Principal grows with rate $r$, compounded $n$ times.

47. What is the perimeter model for a rectangle with length $L$ and width $W$?

    $P=2L+2W$. Sum of all four sides of rectangle.

48. What is the uniform motion model relating distance $d$, rate $r$, and time $t$?

    $d=rt$. Distance equals rate times time at constant speed.

49. What is the formula for simple interest?

    Interest = Principal × Rate × Time. Linear interest calculation over time.

50. What is the volume of a cube with side length 5?

    Volume = 125. Cube volume is $s^3 = 5^3 = 125$.

51. State the equation for the slope of a line given two points $(x_1, y_1)$ and $(x_2, y_2)$.

    Slope = $\frac{y_2 - y_1}{x_2 - x_1}$. Rise over run between two coordinate points.

52. A taxi charges $4 plus $2 per mile; what is the linear model for cost $C$ vs miles $m$?

    $C=2m+4$. Fixed cost plus variable cost per mile.

53. What is the conversion formula from Fahrenheit $F$ to Celsius $C$ for modeling temperature?

    $C=\frac{5}{9}(F-32)$. Subtract 32 then multiply by $\frac{5}{9}$.

54. Calculate the slope for points (2,3) and (4,7).

    Slope = 2. Apply slope formula: $\frac{7-3}{4-2} = 2$.

55. State the formula for the circumference of a circle.

    Circumference = $2\pi r$. Multiply radius by $2\pi$ for circle's perimeter.

56. What is the volume model for a cylinder with radius $r$ and height $h$?

    $V=\pi r^2h$. Base area times height for circular cylinder.

57. What is the formula for the area of a rectangle?

    Area = $\text{length} \times \text{width}$. Multiply the two perpendicular dimensions.

58. Find the x-intercept of the equation $2x - 3 = 0$.

    $x = \frac{3}{2}$. Set $y = 0$ and solve: $2x - 3 = 0$ gives $x = \frac{3}{2}$.

59. Choose the rate of change formula.

    Rate of Change = $\frac{\text{change in value}}{\text{change in time}}$. Shows how a quantity changes per unit time.

60. What is the formula for the surface area of a sphere?

    Surface Area = $4\pi r^2$. Four $\pi$ times radius squared.

61. What is the general form of a linear equation?

    $Ax + By = C$. Standard form with integer coefficients.

62. What is the formula for acceleration?

    Acceleration = $\frac{\text{change in velocity}}{\text{time}}$. Rate of change of velocity with respect to time.

63. State the formula for the volume of a sphere.

    Volume = $\frac{4}{3} \pi r^3$. Four-thirds π times radius cubed.

64. What is the formula to convert Celsius to Fahrenheit?

    F = $\frac{9}{5}$C + 32. Multiply Celsius by $\frac{9}{5}$ then add 32.

65. Determine the equation of a line with slope 3 and y-intercept 5.

    $y = 3x + 5$. Slope-intercept form with $m=3$, $b=5$.

66. Identify the best model type when $y$ is proportional to $x^2$.

    Quadratic model. Square relationship creates parabolic shape.

67. What is the formula for the area of a parallelogram?

    Area = base × height. Same as rectangle when height is perpendicular.

68. Calculate the volume of a cylinder with radius 3 and height 5.

    Volume = $45\text{π}$. Cylinder volume: $\text{π}(3^2)(5) = 45\text{π}$.

69. What is the meaning of the slope $m$ in a linear model $y=mx+b$?

    Rate of change of $y$ per $1$ unit of $x$. The constant rate of change in the relationship.

70. What is the distance formula between two points?

    $d = \text{√}((x_2 - x_1)^2 + (y_2 - y_1)^2)$. Pythagorean theorem applied to coordinate plane.

71. State the formula for compound interest.

    Amount = Principal × (1 + $\frac{r}{n}$)^(nt). Interest compounded $n$ times per year for $t$ years.

72. What is the Pythagorean Theorem?

    $a^2 + b^2 = c^2$. Sum of squares of legs equals hypotenuse squared.

73. Determine the equation of a line parallel to $y=2x+3$.

    $y = 2x + c$. Parallel lines have identical slopes.

74. If $f(x)=3x^2-12x+5$, what is the $x$-coordinate of the vertex for this quadratic model?

    $x=2$. Vertex x-coordinate is $\frac{-b}{2a}=\frac{12}{6}=2$.

75. What is the work model for combined rates if job times are $t_1$ and $t_2$?

    $\frac{1}{t}=\frac{1}{t_1}+\frac{1}{t_2}$. Combined rate is sum of individual rates.

76. Find the surface area of a cube with side length 4.

    Surface Area = 96. Cube surface area: $6s^2 = 6(16) = 96$.

77. Solve for $x$: $2x + 3 = 7$.

    $x = 2$. Subtract 3 from both sides, then divide by 2.

78. What is the exponential growth formula?

    $y = a(1 + r)^t$. Initial value $a$ grows at rate $r$ over time $t$.

79. State the formula for converting Fahrenheit to Celsius.

    C = $\frac{5}{9}$(F - 32). Subtract 32 first, then multiply by $\frac{5}{9}$.

80. What is the formula for the area of a triangle?

    Area = $\frac{1}{2}$ base × height. Half the base times perpendicular height.

81. Find the midpoint of the segment with endpoints (2, 3) and (4, 7).

    $(3, 5)$. Average the x-coordinates and y-coordinates separately.

82. What is the area model for a rectangle with length $L$ and width $W$?

    $A=LW$. Product of length and width.

83. Find the distance between points (3, 4) and (6, 8).

    $5$. Use distance formula: $\sqrt{(6-3)^2 + (8-4)^2} = \sqrt{9+16} = 5$.

84. What is the formula for the perimeter of a rectangle?

    Perimeter = $2(\text{length} + \text{width})$. Sum of all four sides of rectangle.

85. What is the formula for the area of a parallelogram?

    Area = $\text{base} \times \text{height}$. Same as rectangle when sides are perpendicular.

86. What is the volume model for a rectangular prism with dimensions $L$, $W$, and $H$?

    $V=LWH$. Product of all three dimensions.

87. What is the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$?

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Pythagorean theorem applied to coordinate plane.

88. What is the conversion formula from Celsius $C$ to Fahrenheit $F$ for modeling temperature?

    $F=\frac{9}{5}C+32$. Multiply by $\frac{9}{5}$ then add 32.

89. How do you calculate the slope of a line from its equation $y = mx + b$?

    Slope = $m$. Coefficient of $x$ in slope-intercept form.

90. Identify the vertex form of a quadratic equation.

    $y = a(x-h)^2 + k$. Parabola with vertex at $(h,k)$.

91. Convert 45 degrees to radians.

    $\frac{π}{4}$ radians. Multiply by $\frac{π}{180}$ to convert degrees to radians.

92. What is the formula for density?

    Density = mass/volume. Mass per unit volume ratio.

93. What is the area model for a triangle with base $b$ and height $h$?

    $A=\frac{1}{2}bh$. Half the product of base and height.

94. What is the simple interest model for principal $P$ at annual rate $r$ for $t$ years?

    $A=P(1+rt)$. Interest calculated only on principal amount.

95. Identify the formula for converting Celsius to Fahrenheit.

    $F = \frac{9}{5}C + 32$. Multiply by $\frac{9}{5}$ then add 32 degrees.

96. What is the slope-intercept form for a linear model relating $y$ to $x$?

    $y=mx+b$. Linear equation with $m$ as slope and $b$ as y-intercept.

97. What is the formula for the volume of a cylinder?

    Volume = πr²h. Base area times height for cylinders.

98. What is the inverse variation model relating $y$ to $x$ with constant $k$?

    $y=\frac{k}{x}$. When $y$ is proportional to $\frac{1}{x}$.

99. What is the direct variation model relating $y$ to $x$ with constant $k$?

    $y=kx$. When $y$ is proportional to $x$.

100. What is the density model relating mass $m$, density $\rho$, and volume $V$?

     $\rho=\frac{m}{V}$. Mass per unit volume relationship.

101. If $y$ varies directly with $x$ and $y=18$ when $x=6$, what is $k$ in $y=kx$?

     $k=3$. From $18=k(6)$, so $k=3$.

102. What is the formula for the area of a triangle?

     Area = $\frac{1}{2} \times \text{base} \times \text{height}$. Base times height divided by 2 gives triangle area.

103. Convert 300 degrees to radians.

     $\frac{5\pi}{3}$ radians. Multiply by $\frac{\pi}{180}$: $300 × \frac{\pi}{180} = \frac{5\pi}{3}$.

104. Identify the intercept $b$ for the model $y=-3x+7$.

     $b=7$. The constant term in slope-intercept form.

105. Find the circumference of a circle with diameter 10.

     Circumference = $10\text{π}$. Circumference equals π times diameter.

106. Identify the formula for the area of a circle.

     Area = πr². Pi times radius squared for circular area.

107. State the midpoint formula for two points.

     $\text{Midpoint} = \bigg(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\bigg)$. Average the x-coordinates and y-coordinates.

108. Find the slope of the linear model through $(2,5)$ and $(6,1)$.

     $m=-1$. Slope $m=\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

109. What is the equation of a circle in standard form?

     $(x-h)^2 + (y-k)^2 = r^2$. Center $ (h,k) $ with radius $ r $ squared.

110. Solve for $z$: $5z + 2 = 17$.

     $z = 3$. Subtract 2 from both sides, then divide by 5.

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