homework
1. what is the height of this rectangle? (simplified radical)
2. solve for x and y. (simplified radical form)
3. solve for x. (round answer to the nearest hundredth)
4. solve for x. (simplified radical form)
5. find the value of x and y. (simplified radical)
6. solve for x. (round answer to the nearest hundredth)
7. find the value of g and h. (round answers to the nearest hundredth)
8. veronica bought a high def tv that has a 36 in diagonal screen. the tv is 24 inches tall. will the tv fit in a cabinet that is 27.5 inches wide?

Question

Accepted Answer

{ "quality": "clear", "question_count": 8, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Pythagorean Theorem", "30-60-90 Triangle Theorem", "Trigonometric Ratios" ], "new_concepts": [], "current_concepts": [ "Pythagorean Theorem", "30-60-90 Triangle Theorem", "Trigonometric Ratios", "Special Right Triangles" ] } ## Question 1: Solve for height $h$ of the rectangle $$ \begin{aligned} h^2 + 9^2 &= 10^2 \ h^2 + 81 &= 100 \ h^2 &= 19 \ h &= \sqrt{19} \end{aligned} $$ ## Question 2: Solve for $x$ and $y$ in the $30^\circ$-$60^\circ$-$90^\circ$ triangle $$ \begin{aligned} x &= \frac{14}{2} = 7 \ y &= x\sqrt{3} = 7\sqrt{3} \end{aligned} $$ ## Question 3: Solve for $x$ using sine ratio $$ \begin{aligned} \sin(21^\circ) &= \frac{x}{13} \ x &= 13 \cdot \sin(21^\circ) \ x &\approx 13 \cdot 0.358368 \approx 4.66 \end{aligned} $$ ## Question 4: Solve for $x$ in the right triangle $$ \begin{aligned} \sin(50^\circ) &= \frac{x}{15} \ x &= 15 \cdot \sin(50^\circ) \end{aligned} $$ ## Question 5: Solve for $x$ and $y$ in the $45^\circ$-$45^\circ$-$90^\circ$ triangle $$ \begin{aligned} x = y &= \frac{12}{\sqrt{2}} \ x = y &= \frac{12\sqrt{2}}{2} = 6\sqrt{2} \end{aligned} $$ ## Question 6: Solve for $x$ using cosine ratio $$ \begin{aligned} \cos(24^\circ) &= \frac{x}{13} \ x &= 13 \cdot \cos(24^\circ) \ x &\approx 13 \cdot 0.913545 \approx 11.88 \end{aligned} $$ ## Question 7: Solve for $g$ and $h$ using tangent and cosine ratios $$ \begin{aligned} \tan(35^\circ) &= \frac{g}{52} \implies g = 52 \cdot \tan(35^\circ) \approx 52 \cdot 0.700208 \approx 36.41 \ \cos(35^\circ) &= \frac{52}{h} \implies h = \frac{52}{\cos(35^\circ)} \approx \frac{52}{0.819152} \approx 63.48 \end{aligned} $$ ## Question 8: Determine if the TV fits in the cabinet $$ \begin{aligned} w^2 + 24^2 &= 36^2 \ w^2 + 576 &= 1296 \ w^2 &= 720 \ w &= \sqrt{720} \approx 26.83\text{ inches} \ 26.83\text{ inches} &< 27.5\text{ inches} \implies \text{Yes} \end{aligned} $$ ## Question 1 $\sqrt{19}$ ## Question 2 $x = 7$, $y = 7\sqrt{3}$ ## Question 3 $4.66$ ## Question 4 $15\sin(50^\circ)$ ## Question 5 $x = 6\sqrt{2}$, $y = 6\sqrt{2}$ ## Question 6 $11.88$ ## Question 7 $g \approx 36.41$, $h \approx 63.48$ ## Question 8 Yes, because the width of the TV is approximately $26.83$ inches, which is less than $27.5$ inches. { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Trigonometric Ratios" ] }

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QUESTION IMAGE

Question

Explanation:

Question 1: Solve for height \(h\) of the rectangle

Question 2: Solve for \(x\) and \(y\) in the \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle

Question 3: Solve for \(x\) using sine ratio

Question 4: Solve for \(x\) in the right triangle

Question 5: Solve for \(x\) and \(y\) in the \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle

Question 6: Solve for \(x\) using cosine ratio

Question 7: Solve for \(g\) and \(h\) using tangent and cosine ratios

Question 8: Determine if the TV fits in the cabinet

Answer:

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8