ACT Math Flashcards: Mathematical Modeling

What this deck covers

This deck focuses on Mathematical Modeling, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Find the circumference of a circle with diameter 10.

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

Flashcard 2: If $y$ varies inversely with $x$ and $y=5$ when $x=8$, what is $y$ when $x=10$?

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

Flashcard 3: What is the formula for the volume of a cylinder?

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

Flashcard 4: State the formula for compound interest.

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

Flashcard 5: What is the midpoint formula between points $(x_1,y_1)$ and $(x_2,y_2)$?

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

Flashcard 6: What is the formula for the surface area of a sphere?

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

Flashcard 7: Find the distance between points (3, 4) and (6, 8).

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

Flashcard 8: What is the weighted average model for mean of values $x_1,x_2$ with weights $w_1,w_2$?

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

Flashcard 9: What is the slope formula between points $(x_1,y_1)$ and $(x_2,y_2)$?

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

Flashcard 10: What is the formula for the area of a trapezoid?

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

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

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

Flashcard 12: Given $A=50(1.1)^t$, what is $A$ when $t=0$?

Answer: $50$. Initial value when exponent equals zero.

Flashcard 13: What is the formula for the sum of the interior angles of a polygon?

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

Flashcard 14: Find the surface area of a cube with side length 4.

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

Flashcard 15: Find the area of a circle with radius 4.

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

Flashcard 16: What is the formula for the perimeter of a rectangle?

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

Flashcard 17: What is the volume model for a rectangular prism with dimensions $L$, $W$, and $H$?

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

Flashcard 18: Identify the formula for the circumference of a circle.

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

Flashcard 19: What is the Pythagorean Theorem?

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

Flashcard 20: Identify the best model type when $y$ changes by a constant percent per unit increase in $x$.

Answer: Exponential model. Constant percent change indicates exponential relationship.

Flashcard 21: If $y$ varies inversely with $x$ and $y=5$ when $x=8$, what is $y$ when $x=10$?

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

Flashcard 22: A quantity decays $20\%$ per step; write the factor $b$ in $y=a b^x$.

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

Flashcard 23: What is the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$?

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

Flashcard 24: For the exponential model $y=200(1.05)^t$, what is the growth factor per unit $t$?

Answer: $1.05$. The base of the exponential function.

Flashcard 25: Calculate the slope for points (2,3) and (4,7).

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

Flashcard 26: What is the exponential growth/decay model with initial value $a$ and factor $b$?

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

Flashcard 27: What is the quadratic formula?

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

Flashcard 28: What is the formula for the area of a rectangle?

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

Flashcard 29: Calculate the Fahrenheit for 25°C.

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

Flashcard 30: Identify the best model type when the change in $y$ is a constant amount per unit increase in $x$.

Answer: Linear model. Constant rate of change indicates linear relationship.

Flashcard 31: Calculate the volume of a sphere with radius 2.

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

Flashcard 32: What is the density model relating mass $m$, density $\rho$, and volume $V$?

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

Flashcard 33: Identify the best model type when the change in $y$ is a constant amount per unit increase in $x$.

Answer: Linear model. Constant rate of change indicates linear relationship.

Flashcard 34: What is the formula for compound interest?

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

Flashcard 35: What is the formula for the future value of an annuity?

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

Flashcard 36: What is the point-slope form of a line with slope $m$ through $(x_1,y_1)$?

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

Flashcard 37: Identify the vertex form of a quadratic equation.

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

Flashcard 38: Convert 120 degrees to radians.

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

Flashcard 39: How do you find the volume of a rectangular prism?

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

Flashcard 40: State the formula for the Pythagorean theorem.

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

Flashcard 41: What is the meaning of the intercept $b$ in a linear model $y=mx+b$?

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

Flashcard 42: What is the area of a triangle with base 6 and height 4?

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

Flashcard 43: State the quadratic formula.

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

Flashcard 44: What is the slope formula between points $(x_1,y_1)$ and $(x_2,y_2)$?

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

Flashcard 45: State the midpoint formula for two points.

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

Flashcard 46: Identify the formula for the surface area of a cylinder.

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

Flashcard 47: Find the equation of a line with slope 2 through point (1, 3).

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

Flashcard 48: What is the area model for a triangle with base $b$ and height $h$?

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

Flashcard 49: If $y$ varies directly with $x$ and $y=18$ when $x=6$, what is $k$ in $y=kx$?

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

Flashcard 50: Find the area of a rectangle with length 7 and width 3.

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

Flashcard 51: What is the formula for the area of a parallelogram?

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

Flashcard 52: What is the vertex form of a quadratic model with vertex $(h,k)$?

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

Flashcard 53: Identify the best model type when $xy$ is constant for varying $x$ and $y$.

Answer: Inverse variation. Product remains constant as variables change.

Flashcard 54: What is the formula for the area of a sector of a circle?

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

Flashcard 55: Identify the intercept $b$ for the model $y=-3x+7$.

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

Flashcard 56: What is the general form of a linear equation?

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

Flashcard 57: Determine the y-intercept of the equation $y = 3x + 4$.

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

Flashcard 58: Determine the equation of a line with slope 3 and y-intercept 5.

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

Flashcard 59: What is the compound interest model for principal $P$ at rate $r$ compounded $n$ times for $t$ years?

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

Flashcard 60: What is the formula for the perimeter of a rectangle?

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

Flashcard 61: Solve for $y$: $3y - 2 = 10$.

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

Flashcard 62: What is the formula for the area of a triangle?

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

Flashcard 63: What is the direct variation model relating $y$ to $x$ with constant $k$?

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

Flashcard 64: Identify the formula for converting Celsius to Fahrenheit.

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

Flashcard 65: Convert 300 degrees to radians.

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

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

Answer: Quadratic model. Square relationship creates parabolic shape.

Flashcard 67: What is the average rate of change of $f$ from $x=a$ to $x=b$?

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

Flashcard 68: Find the linear model $y=mx+b$ through $(0,3)$ and $(4,11)$.

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

Flashcard 69: Calculate the volume of a sphere with radius 2.

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

Flashcard 70: For the exponential model $y=200(1.05)^t$, what is the growth factor per unit $t$?

Answer: $1.05$. The base of the exponential function.

Flashcard 71: What is the volume of a cube with side length 5?

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

Flashcard 72: What is the average rate of change of $f$ from $x=a$ to $x=b$?

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

Flashcard 73: How do you calculate the slope of a line from its equation $y = mx + b$?

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

Flashcard 74: Identify the best model type when $y$ changes by a constant percent per unit increase in $x$.

Answer: Exponential model. Constant percent change indicates exponential relationship.

Flashcard 75: Find the midpoint of the segment with endpoints (2, 3) and (4, 7).

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

Flashcard 76: Identify the formula for the surface area of a cylinder.

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

Flashcard 77: State the quadratic formula.

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

Flashcard 78: What is the formula for acceleration?

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

Flashcard 79: What is the distance formula between two points?

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

Flashcard 80: Identify the formula for the slope of a line given two points.

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

Flashcard 81: What is the formula for the perimeter of a rectangle?

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

Flashcard 82: What is the formula for the volume of a sphere?

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

Flashcard 83: What is the uniform motion model relating distance $d$, rate $r$, and time $t$?

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

Flashcard 84: What is the formula for the volume of a cone?

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

Flashcard 85: What is the formula for the volume of a cylinder?

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

Flashcard 86: What is the formula for the area of a rectangle?

Answer: Area = length × width. Base formula for calculating space inside a rectangle.

Flashcard 87: State the formula for the Pythagorean theorem.

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

Flashcard 88: What is the compound interest model for principal $P$ at rate $r$ compounded $n$ times for $t$ years?

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

Flashcard 89: What is the formula to convert Celsius to Fahrenheit?

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

Flashcard 90: What is the formula for the future value of an annuity?

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

Flashcard 91: What is the standard form of a quadratic model in $x$?

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

Flashcard 92: What is the formula for compound interest?

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

Flashcard 93: What is the perimeter model for a rectangle with length $L$ and width $W$?

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

Flashcard 94: What is the slope-intercept form for a linear model relating $y$ to $x$?

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

Flashcard 95: Find the linear model $y=mx+b$ through $(0,3)$ and $(4,11)$.

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

Flashcard 96: State the formula for the volume of a sphere.

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

Flashcard 97: If $f(x)=3x^2-12x+5$, what is the $x$-coordinate of the vertex for this quadratic model?

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

Flashcard 98: Find the area of a circle with radius 4.

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

Flashcard 99: What is the simple interest model for principal $P$ at annual rate $r$ for $t$ years?

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

Flashcard 100: What is the perimeter model for a rectangle with length $L$ and width $W$?

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