Grade 11 General Mathematics: Solid Geometry, Unit Conversions, and Functions

Comprehensive review practice flashcards covering solid geometry (surface area and volume formulas and applications), unit conversions (scale drawings, international currencies, time intervals, and temperature scales), and algebraic functions (linear, quadratic, absolute value, and radical functions).

What is the definition of the surface area of a solid figure?

Surface area is the total area of all flat or curved surfaces that cover the outside of a three-dimensional solid figure, measured in square units.

What is the formula for calculating the surface area of a rectangular prism?

$SA = 2(lh + lw + hw)$, where $l$ is the length, $h$ is the height, and $w$ is the width.

What is the surface area of the rectangular prism shown in the figure?

$1032\,\text{cm}^2$. Calculated as $SA = 2[(18 \times 12) + (18 \times 10) + (12 \times 10)] = 2(216 + 180 + 120) = 2(516) = 1032\,\text{cm}^2$.

What is the general formula for finding the surface area of any pyramid?

$$SA = \text{area of the base} + \text{sum of areas of all triangular faces}$$

What is the formula for the total surface area of a cylinder?

$SA = 2\pi r^2 + 2\pi rh$, where $r$ is the radius of the circular base and $h$ is the height.

What is the formula for the total surface area of a cone?

$SA = \pi r^2 + \pi rl$, where $r$ is the radius of the circular base and $l$ is the slant height.

What is the formula for the surface area of a sphere, and what is the surface area of a sphere with radius $r = 7\,\text{cm}$ using $\pi = \frac{22}{7}$?

The formula is $SA = 4\pi r^2$. For $r = 7\,\text{cm}$, $SA = 4\left(\frac{22}{7}\right)(7^2) = 616\,\text{cm}^2$.

What is the definition of the volume of a solid figure?

Volume is the amount of three-dimensional space an object occupies, indicating how much substance (liquid, gas, or solid) can fit inside a given shape, expressed in cubic units such as $\text{cm}^3$, $\text{m}^3$, or $\text{L}$.

What is the formula for the volume of a cube, and what is the volume of a cube with side length $s = 12\,\text{cm}$?

The formula is $V = s^3$. For $s = 12\,\text{cm}$, $V = 12^3 = 1728\,\text{cm}^3$.

What is the general formula for calculating the volume of any prism?

$V = A_{\text{base}} \times h$, where $A_{\text{base}}$ is the area of the base and $h$ is the height or distance between the two parallel bases.

What is the general formula for calculating the volume of any pyramid?

$V = \frac{1}{3} \times A_{\text{base}} \times h$, where $A_{\text{base}}$ is the area of the base and $h$ is the perpendicular height from the vertex to the base.

What is the volume formula for a cylinder, and what is its value when $r = 3\,\text{cm}$ and $h = 8\,\text{cm}$ (using $\pi \approx 3.14$)?

The formula is $V = \pi r^2 h$. Substituting the values gives $V = (3.14)(3^2)(8) \approx 226.08\,\text{cm}^3$.

How does the volume formula of a cone relate to the volume formula of a cylinder with the same base radius and height?

The volume of a cone is given by $V = \frac{1}{3}\pi r^2 h$, which is exactly one-third the volume of a cylinder with the identical base radius and height.

What is the formula for the volume of a sphere, and what is the volume when $r = 10\,\text{cm}$ (using $\pi \approx 3.14$)?

The formula is $V = \frac{4}{3}\pi r^3$. For $r = 10\,\text{cm}$, $V = \frac{4}{3}(3.14)(10^3) \approx 4186.67\,\text{cm}^3$.

How is the total volume of a composite figure found, such as a monument with a rectangular prism base ($2\,\text{m} \times 2\,\text{m} \times 4\,\text{m}$) topped by a square pyramid of height $3\,\text{m}$?

Calculate the volume of each part separately and sum them: $V_{\text{prism}} = 2 \times 2 \times 4 = 16\,\text{m}^3$ and $V_{\text{pyramid}} = \frac{1}{3}(2^2)(3) = 4\,\text{m}^3$, yielding a total volume of $16 + 4 = 20\,\text{m}^3$.

In practical applications, what distinct physical properties do surface area and volume measure?

Surface area determines the amount of material needed to cover an exterior (e.g., paint, metal coating, or packaging), whereas volume measures how much three-dimensional space is enclosed or how much content a container can hold.

What are the six structured steps for solving surface area and volume word problems?

1. Understand the problem (identify given and asked values); 2. Visualize and identify the solid figure and dimensions; 3. Choose the correct formula; 4. Perform step-by-step calculations; 5. Use proper units; 6. Check work for completeness and accuracy.

What is the volume of a rectangular steel beam with length $8\,\text{m}$, width $0.3\,\text{m}$, and height $0.5\,\text{m}$?

$1.2\,\text{m}^3$. Calculated as $V = l \times w \times h = 8 \times 0.3 \times 0.5 = 1.2\,\text{m}^3$.

What is a scale in technical drawings and maps?

A scale shows the proportional relationship between a measurement on a drawing or model and the actual real-world measurement it represents.

What is a scale factor, and what is the scale factor for a map scale of $2\,\text{cm} : 5\,\text{km}$?

A scale factor is the number used to multiply or divide drawing measurements to obtain actual dimensions (or vice versa). For $2\,\text{cm} : 5\,\text{km}$, the scale factor is $5 \div 2 = 2.5\,\text{km/cm}$.

What is an exchange rate?

An exchange rate specifies how much one unit of a particular currency is worth in terms of another currency.

According to the Bangko Sentral ng Pilipinas rates as of June 25, 2025, what are the conversions for $1\,\text{USD}$, $1\,\text{JPY}$, and $1\,\text{GBP}$ in Philippine Pesos ($\text{PHP}$)?

$1\,\text{USD} = 56.9870\,\text{PHP}$, $1\,\text{JPY} = 0.3934\,\text{PHP}$, and $1\,\text{GBP} = 77.5992\,\text{PHP}$.

How do you convert $1000\,\text{PHP}$ to US dollars using the exchange rate $1\,\text{USD} = 56.9870\,\text{PHP}$?

Divide the peso amount by the exchange rate: $1000\,\text{PHP} \cdot \frac{1\,\text{USD}}{56.9870\,\text{PHP}} \approx 17.55\,\text{USD}$.

What are the standard conversion ratios among days, hours, minutes, and seconds?

$1\,\text{day} = 24\,\text{hours}$, $1\,\text{hour} = 60\,\text{minutes}$, and $1\,\text{minute} = 60\,\text{seconds}$.

How many minutes are in $4\,\text{days}$?

$5760\,\text{minutes}$. Calculated as $4\,\text{days} \times \frac{24\,\text{hours}}{1\,\text{day}} \times \frac{60\,\text{minutes}}{1\,\text{hour}} = 5760\,\text{minutes}$.

How many total days are there between March 24, 2025, and May 15, 2025?

$52\,\text{days}$. Calculated as: March has $31 - 24 = 7\,\text{days}$, April has $30\,\text{days}$, and May has $15\,\text{days}$, giving $7 + 30 + 15 = 52\,\text{days}$.

What is a linear function, and what is its slope-intercept form?

A linear function models a constant rate of change between two variables, forming a straight line. Its slope-intercept form is $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

What are the point-slope form and the standard form of a linear equation?

Point-slope form is $y - y_1 = m(x - x_1)$, and standard form is $Ax + By = C$, where $A$, $B$, and $C$ are real numbers and $A$ and $B$ are not both zero.

What are the domain and range of a linear function $f(x) = mx + b$ where $m \neq 0$?

Both the domain and range are the set of all real numbers, expressed as $(-\infty, \infty)$.

What algebraic indicators confirm that an equation represents a linear function?

The variable $x$ must have an exponent of 1, and the equation must not contain products of variables, roots of variables, absolute values, or higher-degree powers of $x$.

What are the formulas to convert between Celsius ($^{\circ}\text{C}$) and Fahrenheit ($^{\circ}\text{F}$)?

$^{\circ}\text{F} = {^{\circ}\text{C}} \cdot \frac{9}{5} + 32$ and $^{\circ}\text{C} = ({^{\circ}\text{F}} - 32) \cdot \frac{5}{9}$.

What are the formulas to convert between Celsius ($^{\circ}\text{C}$) and Kelvin ($\text{K}$)?

$\text{K} = {^{\circ}\text{C}} + 273.15$ and $^{\circ}\text{C} = \text{K} - 273.15$.

What is the formula to convert Fahrenheit ($^{\circ}\text{F}$) to Kelvin ($\text{K}$), and what is $98^{\circ}\text{F}$ in Kelvin?

The formula is $\text{K} = ({^{\circ}\text{F}} - 32) \cdot \frac{5}{9} + 273.15$. For $98^{\circ}\text{F}$, $\text{K} = (98 - 32) \cdot \frac{5}{9} + 273.15 = 36.67 + 273.15 = 309.82\,\text{K}$.

What is a quadratic function, and what geometric curve does its graph produce?

A quadratic function is a second-degree polynomial function written as $f(x) = ax^2 + bx + c$ where $a \neq 0$. Its graph is a U-shaped curve called a parabola.

What are the three common algebraic forms of a quadratic function?

Standard form: $f(x) = ax^2 + bx + c$; Vertex form: $f(x) = a(x - h)^2 + k$; Intercepts form: $f(x) = a(x - r_1)(x - r_2)$.

How are the vertex coordinates $(h, k)$ of a parabola calculated directly from the standard form $f(x) = ax^2 + bx + c$?

$h = \frac{-b}{2a}$ and $k = \frac{4ac - b^2}{4a}$.

For the quadratic function $f(x) = x^2 - 6x + 5$, what are the vertex, axis of symmetry, and range?

In vertex form, $f(x) = (x - 3)^2 - 4$. The vertex is $(3, -4)$, the axis of symmetry is $x = 3$, and the range is $[-4, \infty)$.

What is an absolute value function, and what is its standard vertex form?

An absolute value function is a piecewise-defined function containing an absolute value expression. Its general vertex form is $f(x) = a|x - h| + k$, where $(h, k)$ is the vertex.

How is the parent absolute value function $f(x) = |x|$ defined piecewise?

$f(x) = x$ if $x \ge 0$, and $f(x) = -x$ if $x < 0$.

What are the domain and range of the absolute value function $f(x) = a|x - h| + k$?

The domain is always $(-\infty, \infty)$. The range is $[k, \infty)$ if $a > 0$ (opens upward) and $(-\infty, k]$ if $a < 0$ (opens downward).

What are the vertex, axis of symmetry, and direction of opening for the absolute value function shown in the graph?

Vertex: $(-2, -1)$; Axis of symmetry: $x = -2$; Direction of opening: upward (with a minimum value of $-1$), corresponding to $f(x) = |x + 2| - 1$.

What is a radical function, and what defines square root and cube root functions?

A radical function is a function defined by a radical expression containing the variable inside the radical sign. A square root function has an index of 2, while a cube root function has an index of 3.

How does an odd versus an even index affect the domain and range of a radical function?

For an odd index, the radicand can be positive, negative, or zero, so both the domain and range are $(-\infty, \infty)$. For an even index, the radicand cannot be negative ($\ge 0$), restricting the domain and range.

How is the radical function $f(x) = \sqrt{2x + 1}$ evaluated when $x = \frac{5}{2}$?

Substitute $x = \frac{5}{2}$: $f\left(\frac{5}{2}\right) = \sqrt{2\left(\frac{5}{2}\right) + 1} = \sqrt{5 + 1} = \sqrt{6}$.

What are the domain and range of the radical function $f(x) = -\sqrt{x} + 2$?

Domain: $x \ge 0$ or $[0, \infty)$, because the index is even and the radicand cannot be negative. Range: $f(x) \le 2$ or $(-\infty, 2]$, because $-\sqrt{x} \le 0$.

What is the total surface area of a rectangular pyramid with a base measuring $6\,\text{m} \times 4\,\text{m}$ and triangular face slant heights of $9\,\text{m}$ and $8\,\text{m}$?

$110\,\text{m}^2$. The base area is $6 \times 4 = 24\,\text{m}^2$. The areas of the two pairs of triangular faces are $2 \times \left(\frac{1}{2} \times 6 \times 9\right) = 54\,\text{m}^2$ and $2 \times \left(\frac{1}{2} \times 4 \times 8\right) = 32\,\text{m}^2$. Adding the base and lateral faces gives $24 + 54 + 32 = 110\,\text{m}^2$.

How much colored paper is needed to completely cover a cone-shaped party hat (including the circular base) with radius $5\,\text{cm}$ and slant height $13\,\text{cm}$?

$282.6\,\text{cm}^2$. Using the total surface area formula $SA = \pi r^2 + \pi rl$, substituting the dimensions yields $SA = (3.14)(5^2) + (3.14)(5)(13) = 78.5 + 204.1 = 282.6\,\text{cm}^2$.

For a rectangular steel beam measuring $8\,\text{m} \times 0.3\,\text{m} \times 0.5\,\text{m}$, what surface area is coated if the bottom face ($8\,\text{m} \times 0.3\,\text{m}$) is excluded?

$10.7\,\text{m}^2$. The top face is $8 \times 0.3 = 2.4\,\text{m}^2$, the two side faces are $2(8 \times 0.5) = 8.0\,\text{m}^2$, and the two end faces are $2(0.3 \times 0.5) = 0.3\,\text{m}^2$. Summing these gives $2.4 + 8.0 + 0.3 = 10.7\,\text{m}^2$.

A cylindrical mousse container has a volume of $502.4\,\text{cm}^3$ and a height of $10\,\text{cm}$. What is the radius of the container (using $\pi \approx 3.14$)?

$4\,\text{cm}$. Using the formula $V = \pi r^2 h$, substituting gives $502.4 = (3.14)r^2(10)$, which simplifies to $502.4 = 31.4r^2$, resulting in $r^2 = 16$ and $r = 4\,\text{cm}$.

If a scale drawing uses a scale of $3\,\text{cm} : 8\,\text{m}$, what is the blueprint length of an actual fence that is $5\,\text{m}$ long?

$1.875\,\text{cm}$. Setting up the proportion $\frac{3\,\text{cm}}{8\,\text{m}} = \frac{x}{5\,\text{m}}$ gives $x = \frac{3 \times 5}{8} = 1.875\,\text{cm}$.

A floor plan uses a scale of $3\,\text{cm} : 10\,\text{m}$. What is the area on the floor plan of a living room that actually measures $8\,\text{m} \times 6\,\text{m}$?

$4.32\,\text{cm}^2$. The drawing dimensions are $8 \times \left(\frac{3}{10}\right) = 2.4\,\text{cm}$ and $6 \times \left(\frac{3}{10}\right) = 1.8\,\text{cm}$. The area on the plan is $2.4 \times 1.8 = 4.32\,\text{cm}^2$.

A purse costs $82000\,\text{JPY}$ in Japan ($1\,\text{JPY} = 0.3934\,\text{PHP}$) and $770000\,\text{KRW}$ in South Korea ($1\,\text{KRW} = 0.0420\,\text{PHP}$). Which country offers the cheaper price?

Japan offers the cheaper price. In Japan, $82000 \times 0.3934 = 32258.80\,\text{PHP}$, whereas in South Korea, $770000 \times 0.0420 = 32340.00\,\text{PHP}$.

A printing shop charges costs modeled by $P(x) = 2x + 500$, where $x$ is pages printed. What is the total cost for $150\,\text{pages}$, and what are the domain and range for up to $300\,\text{pages}$?

Total cost: $800\,\text{PHP}$, evaluated as $P(150) = 2(150) + 500 = 800\,\text{PHP}$. The domain is $[0, 300]$ and the range is $[500, 1100]$.

What is the equivalent temperature in Celsius for a hot oven set at $400^{\circ}\text{F}$?

$204.44^{\circ}\text{C}$. Using the conversion formula $^{\circ}\text{C} = ({^{\circ}\text{F}} - 32) \times \frac{5}{9}$, substituting yields $^{\circ}\text{C} = (400 - 32) \times \frac{5}{9} = 368 \times \frac{5}{9} \approx 204.44^{\circ}\text{C}$.

For the profit function $P(x) = -10x^2 + 500x - 3000$ (where $x$ is tickets sold in hundreds), what number of tickets maximizes profit, and what is the maximum profit in pesos?

$2500\,\text{tickets}$ and a maximum profit of $3250\,\text{PHP}$. The vertex $x$-coordinate is $x = \frac{-500}{2(-10)} = 25$ (representing $2500\,\text{tickets}$). The maximum profit is $P(25) = -10(25^2) + 500(25) - 3000 = 3250\,\text{PHP}$.

A taxi fare is modeled by $f(x) = 30|x + 2| + 50$, where $x$ is passenger position in kilometers relative to the center. What is the fare for a passenger located $5\,\text{km}$ west ($x = -5$)?

$140\,\text{PHP}$. Substituting $x = -5$ gives $f(-5) = 30|(-5) + 2| + 50 = 30|-3| + 50 = 30(3) + 50 = 140\,\text{PHP}$.

For the radical function $f(x) = \frac{3\sqrt{x + 4}}{2}$, what is the value of $f(12)$?

$6$. Substituting $x = 12$ gives $f(12) = \frac{3\sqrt{12 + 4}}{2} = \frac{3\sqrt{16}}{2} = \frac{3(4)}{2} = 6$.

What are the domain and range of the radical function $f(x) = \sqrt[3]{2x - 5} - 2$?

Both the domain and range are all real numbers, expressed as $(-\infty, \infty)$, because the radical index is odd ($3$), which allows the radicand and output values to be positive, negative, or zero.

What is the total surface area of a pyramid with a square base of side length $3\,\text{m}$ and triangular faces with slant height $5\,\text{m}$?

$39\,\text{m}^2$. The base area is $s^2 = 3^2 = 9\,\text{m}^2$. The lateral area of the four triangular faces is $4 \times \left(\frac{1}{2} \times 3 \times 5\right) = 4 \times 7.5 = 30\,\text{m}^2$. Adding the base and lateral area yields $9 + 30 = 39\,\text{m}^2$.

What is the volume of a rectangular pyramid with a base measuring $6\,\text{m} \times 5\,\text{m}$ and a perpendicular height of $8\,\text{m}$?

$80\,\text{m}^3$. Using $V = \frac{1}{3} A<em>{\text{base}} h$, find the base area $A</em>{\text{base}} = 6 \times 5 = 30\,\text{m}^2$. Then, $V = \frac{1}{3}(30)(8) = 80\,\text{m}^3$.

What is the volume of a cone with base radius $r = 5\,\text{cm}$ and height $h = 15\,\text{cm}$ (using $\pi \approx 3.14$)?

$392.5\,\text{cm}^3$. Using the formula $V = \frac{1}{3}\pi r^2 h$, evaluate $V = \frac{1}{3}(3.14)(5^2)(15) = \frac{1}{3}(3.14)(25)(15) = 392.5\,\text{cm}^3$.

What is the total surface area of a cylinder with base radius $r = 6\,\text{cm}$ and height $h = 15\,\text{cm}$ (using $\pi \approx 3.14$)?

$791.28\,\text{cm}^2$. Using $SA = 2\pi r^2 + 2\pi rh$, the combined base area is $2(3.14)(6^2) = 226.08\,\text{cm}^2$ and the lateral area is $2(3.14)(6)(15) = 565.20\,\text{cm}^2$. Summing both gives $226.08 + 565.20 = 791.28\,\text{cm}^2$.

A cylindrical container has a radius of $4\,\text{cm}$ and a height of $10\,\text{cm}$. How much foil is needed to cover only its curved lateral surface (using $\pi \approx 3.14$)?

$251.2\,\text{cm}^2$. Using the lateral area formula $LA = 2\pi rh$, substitute the dimensions: $LA = 2(3.14)(4)(10) = 251.2\,\text{cm}^2$.

If the production cost in hundreds of pesos is modeled by $f(x) = 5x + 2$, what is the total cost to produce $3$ items?

$1700\,\text{PHP}$. Evaluating at $x = 3$ gives $f(3) = 5(3) + 2 = 17$. Because the cost function is in hundreds of pesos, the final cost is $17 \times 100 = 1700\,\text{PHP}$.

For the absolute value function $f(x) = 2|x - 1| + 3$, what are the vertex, direction of opening, domain, and range?

Vertex: $(1, 3)$; Direction of opening: upward (since $a = 2 > 0$); Domain: $(-\infty, \infty)$; Range: $[3, \infty)$.

For the quadratic function $f(x) = x^2 - 6x + 5$, what are the y-intercept and the x-intercepts?

The y-intercept is $(0, 5)$, found by computing $f(0) = 0^2 - 6(0) + 5$. The x-intercepts are $(1, 0)$ and $(5, 0)$, found by solving $(x - 1)(x - 5) = 0$.

What are the values of $f(0)$, $f(1)$, and $f(4)$ for the radical function $f(x) = -\sqrt{x} + 2$?

$f(0) = 2$, $f(1) = 1$, and $f(4) = 0$. Evaluated step-by-step: $f(0) = -\sqrt{0} + 2 = 2$; $f(1) = -\sqrt{1} + 2 = -1 + 2 = 1$; and $f(4) = -\sqrt{4} + 2 = -2 + 2 = 0$.

What is the total surface area of a cone with radius $r = 4\,\text{dm}$ and slant height $l = 6\,\text{dm}$ (using $\pi \approx 3.14$)?

$$125.60\,\text{dm}^2$$

What is the volume of a rectangular prism with length $l = 6\,\text{cm}$, width $w = 4\,\text{cm}$, and height $h = 10\,\text{cm}$?

$$240\,\text{cm}^3$$

What is the volume of a triangular pyramid with a base measuring $5\,\text{cm}$ by $7\,\text{cm}$ and a height of $10\,\text{cm}$?

$$58.33\,\text{cm}^3$$

What is the area of a circular surface with radius $r = 4\,\text{cm}$ (using $\pi \approx 3.14$)?

$$50.24\,\text{cm}^2$$

If a blueprint uses a scale of $4\,\text{cm} : 15\,\text{m}$, what is the blueprint length of a wall that is actually $21\,\text{m}$ long?

$$5.6\,\text{cm}$$

What is $1500\,\text{PHP}$ converted to Singaporean dollars ($\text{SGD}$) using the exchange rate $1\,\text{SGD} = 44.5315\,\text{PHP}$?

$$33.68\,\text{SGD}$$

How many months are in $50\,\text{days}$ using the standard ratio $30\,\text{days} = 1\,\text{month}$?

$$1.67\,\text{months}$$

How many minutes are in $10\,\text{years}$ (assuming $1\,\text{year} = 365\,\text{days}$)?

$$5256000\,\text{minutes}$$

If an individual journals for $1\,\text{hour}$ and $30\,\text{minutes}$ each week for one full year ($52\,\text{weeks}$), how many total days do they spend journaling?

$$3.25\,\text{days}$$

What is $70^{\circ}\text{F}$ converted to Celsius ($^{\circ}\text{C}$)?

$$21.11^{\circ}\text{C}$$

What is $50^{\circ}\text{C}$ converted to Kelvin ($\text{K}$)?

$$323.15\,\text{K}$$

What is $212\,\text{K}$ converted to Celsius ($^{\circ}\text{C}$)?

$$-61.15^{\circ}\text{C}$$

What is $16^{\circ}\text{C}$ converted to Fahrenheit ($^{\circ}\text{F}$)?

$$60.8^{\circ}\text{F}$$

For the linear function $f(x) = 2x - 1$, what is the output value when $x = 2$?

$$3$$

What is the vertex of the quadratic function $f(x) = (x + 6)^2 - 5$?

$$(-6, -5)$$

What is the vertex of the absolute value function $t(x) = 3|x - 2| + 4$?

$$(2, 4)$$

For the absolute value function $f(x) = |x + 2| - 1$, what is the output value when $x = -6$?

$$3$$

For the radical function $f(x) = \sqrt[3]{x - 4}$, what is the value of $f(12)$?

$$2$$