Expressions, Equations, and Relationships
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Texas 7th Grade Math › Expressions, Equations, and Relationships
A triangular prism has base area 30 cm² and height 12 cm, giving volume 360 cm³. A triangular pyramid with the same base and the same height has volume 120 cm³.
What is the relationship between these volumes when the prism and pyramid share congruent bases and heights?
They are always equal if the base and height match.
The pyramid's volume is 1/3 of the prism's volume.
The pyramid's volume is 3 times the prism's volume.
The prism uses $V=\frac{1}{3}Bh$ while the pyramid uses $V=Bh$.
Explanation
For congruent bases and heights, $V_{\text{prism}}=Bh$ and $V_{\text{pyramid}}=\tfrac{1}{3}Bh$, so the pyramid has $\tfrac{1}{3}$ the volume of the prism. Geometrically, three identical pyramids with the same base and height can fill the prism exactly.
A figure consists of a rectangle (12 ft × 8 ft) with a semicircle attached to the 12 ft side. Use 3.14 for π. What is the total area of the composite figure? Round to the nearest tenth.
209.0 square feet
322.1 square feet
152.5 square feet
96.0 square feet
Explanation
Break into a rectangle and a semicircle. Rectangle: A=lw=12×8=96. Semicircle radius is r=6, so A=½πr²=½·3.14·6²=56.52. Total =96+56.52=152.52≈152.5 square feet. Distractors: adding a full circle (96+113.04), using r=12, or omitting the semicircle.
Which equation represents this table?
x: -1, 0, 1, 3 y: -5, -3, -1, 3
$y = 2x + 3$
$y = x - 3$
$y = -2x - 3$
$y = 2x - 3$
Explanation
From the table, as $x$ increases by 1, $y$ increases by 2, so $m=2$. When $x=0$, $y=-3$, so $b=-3$. The equation is $y=2x-3$. Check: for $x=3$, $y=2(3)-3=3$, matching the table.
Which real-world scenario matches the equation $3x + 7 = 22$?
A gym charges 7 dollars per visit plus a 3-dollar membership fee. You paid 22 dollars total. How many visits $x$ did you make?
A phone case costs 3 dollars, and sales tax is 7 dollars. The total is 22 dollars.
A ride costs 3 dollars per mile, plus a 7-dollar pickup fee. The total was 22 dollars. How many miles $x$ did you travel?
You had 3 dollars and then saved 7 dollars each day for $x$ days to reach 22 dollars.
Explanation
The equation $3x + 7 = 22$ means 3 dollars times the number of miles ($x$) plus a flat 7-dollar fee equals a total of 22 dollars. Choice C states exactly that structure: total cost = (3 per mile)$\times x$ + 7, and it equals 22.
A circle has a radius of 7.5 cm. What is the circumference?
47.10 cm
23.55 cm
176.63 square centimeters
94.20 cm
Explanation
Use $C=2\pi r$ because the radius is given. $C=2(3.14)(7.5)=47.1\text{ cm}$ (about $47.10\text{ cm}$). Circumference uses linear units (cm), not square units.
Solve: $5x - 8 = 27$. Which value of $x$ makes this equation true?
5
6
7
8
Explanation
Add 8 to both sides to undo the subtraction: $5x = 35$. Then divide both sides by 5: $x = 7$. Check: $5(7) - 8 = 35 - 8 = 27$, which is true.
A triangular prism has a triangular base with base 9 cm and height 8 cm. The length of the prism is 12 cm. What is the volume?
864 cm³
432 cm³
144 cm³
54 cm³
Explanation
Prisms use $V = Bh$, where $B$ is the area of the base. The base is a triangle, so $B = \tfrac{1}{2}bh = \tfrac{1}{2}\cdot 9\cdot 8 = 36,\text{cm}^2$. Then $V = Bh = 36\cdot 12 = 432,\text{cm}^3$. Distractors: $864$ (forgot $\tfrac{1}{2}$), $144$ (incorrectly divided by 3 as if a pyramid), $54$ (used wrong dimensions).
A music app charges a \$10 monthly fee plus \$1.25 per song download. Jordan's budget allows at most \$35 this month. Let $s$ be the number of songs. What inequality represents this constraint?
$10 + 1.25s \ge 35$
$1.25 + 10s \le 35$
$10 + 1.25s \le 35$
$1.25s = 35$
Explanation
The fixed amount (constant) is $10$ and the rate (coefficient) is \$1.25$ per song, so the expression is $10 + 1.25s$. "At most \$35" means the total cannot exceed 35, so use $\le$: \$10 + 1.25s \le 35$.
Which value(s) make this equation true? Test $x = 4$, $x = 5$, $x = 3$ in $3x - 5 = 7$.
$x = 4$ only
$x = 5$ only
$x = 3$ only
$x = 4$ and $x = 5$
Explanation
Test $x=4$: $3(4)-5=12-5=7$, which equals $7$ (true). Test $x=5$: $3(5)-5=15-5=10$, which is not $7$ (false). Test $x=3$: $3(3)-5=9-5=4$, which is not $7$ (false). Therefore, only $x=4$ makes the equation true.
The angles in a triangle measure $x^\circ$, $(2x + 15)^\circ$, and $(x - 5)^\circ$. Which equation can you write using the triangle angle-sum relationship?
$x + (2x + 15) = (x - 5)$
$x + (2x + 15) + (x - 5) = 180$
$x + (2x + 15) + (x - 5) = 90$
$(2x + 15) = (x - 5)$
Explanation
Triangle angle sum: $180^\circ$. Set up $x + (2x + 15) + (x - 5) = 180$. Simplify: $4x + 10 = 180 \Rightarrow 4x = 170 \Rightarrow x = 42.5$. The angles are $42.5^\circ$, $100^\circ$, and $37.5^\circ$, which add to $180^\circ$. Distractors either set angles equal or use $90^\circ$ incorrectly.