Answer :
Sure, let's work through each instruction and question step-by-step.
Instruction III: Fill in the blank space with the correct answer on the space provided.
21. The simplified form of [tex]\(\frac{\sqrt[3]{375}-4 \sqrt[3]{24}}{\sqrt{61}}\)[/tex] is [tex]\(\frac{\sqrt[3]{375}-4 \sqrt[3]{24}}{\sqrt{61}}\)[/tex].
22. The region bounded by a chord and an arc of a circle is called a _segment_.
23. The simplified form of [tex]\(\frac{\cos 130^{\circ}}{\cos 40^{\circ}} \cdot \frac{1}{\cos 25^{\circ}}\)[/tex] is:
Since [tex]\(\cos 130^\circ = -\cos 50^\circ\)[/tex] and using the complementary angle identity, the expression becomes:
[tex]\[ \frac{-\cos 50^\circ}{\cos 40^\circ} \cdot \frac{1}{\cos 25^\circ} = \frac{-\cos (90^\circ - 40^\circ)}{\cos 40^\circ} \cdot \frac{1}{\cos 25^\circ} = \frac{-\sin 40^\circ}{\cos 40^\circ} \cdot \frac{1}{\cos 25^\circ} \][/tex]
Using a known trigonometric identity, the final simplified form is:
[tex]\[ \boxed{-\tan 40^\circ \cdot \sec 25^\circ} \][/tex]
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Instruction IV: Work out the following question showing all the necessary steps.
24. If the measures of all four interior angles of a cyclic quadrilateral add up to 360 degrees and are given as [tex]\((3x-15)^{\circ}, (2x+10)^{\circ}, (4x-8)^{\circ},\)[/tex] and [tex]\((6x-17)^{\circ}\)[/tex], solve for [tex]\(x\)[/tex]:
- Sum the angles: [tex]\((3x - 15) + (2x + 10) + (4x - 8) + (6x - 17) = 360\)[/tex]
- Simplify by combining like terms and solving for [tex]\(x\)[/tex]:
[tex]\[ 3x + 2x + 4x + 6x - 15 + 10 - 8 - 17 = 360 \][/tex]
[tex]\[ 15x - 30 = 360 \][/tex]
[tex]\[ 15x = 390 \][/tex]
[tex]\[ x = 26 \][/tex]
25. Solve the quadratic equation [tex]\(2x^2 - 6x + 8 = 0\)[/tex]:
Let's rewrite and solve it:
[tex]\[ 2x^2 - 6x + 8 = 0 \][/tex]
Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 2\)[/tex], [tex]\(b = -6\)[/tex], and [tex]\(c = 8\)[/tex]:
[tex]\[ b^2 - 4ac = (-6)^2 - 4(2)(8) = 36 - 64 = -28 \][/tex]
Since the discriminant ([tex]\(\Delta\)[/tex]) is [tex]\(\boxed{-28}\)[/tex] the equation has complex solutions.
26. A ladder leaning against a wall forms an angle of [tex]\(30^{\circ}\)[/tex] with the wall. If the foot of the ladder is 8 m from the wall, find the length of the ladder [tex]\(L\)[/tex]:
Using trigonometry, specifically the sine function:
[tex]\(\sin(30^\circ) = \frac{8}{L}\)[/tex], where [tex]\(\sin 30^\circ = 0.5\)[/tex]:
[tex]\[ 0.5 = \frac{8}{L} \][/tex]
[tex]\[ L = 16 \, \text{meters} \][/tex]
27. Two chords [tex]\(AB\)[/tex] and [tex]\(CD\)[/tex] of a circle intersect at right angles at a point inside the circle. If [tex]\(\angle BAC = 35^\circ\)[/tex], find [tex]\(\angle ABD\)[/tex]:
Since the angles formed by intersecting chords (perpendicular) sum up to [tex]\(90^\circ\)[/tex]:
[tex]\[ \angle BAC + \angle ABD = 90^\circ \][/tex]
[tex]\[ 35^\circ + \angle ABD = 90^\circ \][/tex]
[tex]\[ \angle ABD = 55^\circ \][/tex]
28. Solve the equation [tex]\(2x^2 + x - 1 = 0\)[/tex] in the interval [tex]\([0, 2]\)[/tex]:
Given the discriminant and solutions:
[tex]\[ D = 9, x_1 = 0.5, x_2 = -1.0, \quad \text{solutions in } [0, 2] = [0.5] \][/tex]
Therefore, the solution set for the equation in the interval [tex]\([0, 2]\)[/tex] is:
[tex]\[ \boxed{[0.5]} \][/tex]
This completes the detailed steps for each given question.
Instruction III: Fill in the blank space with the correct answer on the space provided.
21. The simplified form of [tex]\(\frac{\sqrt[3]{375}-4 \sqrt[3]{24}}{\sqrt{61}}\)[/tex] is [tex]\(\frac{\sqrt[3]{375}-4 \sqrt[3]{24}}{\sqrt{61}}\)[/tex].
22. The region bounded by a chord and an arc of a circle is called a _segment_.
23. The simplified form of [tex]\(\frac{\cos 130^{\circ}}{\cos 40^{\circ}} \cdot \frac{1}{\cos 25^{\circ}}\)[/tex] is:
Since [tex]\(\cos 130^\circ = -\cos 50^\circ\)[/tex] and using the complementary angle identity, the expression becomes:
[tex]\[ \frac{-\cos 50^\circ}{\cos 40^\circ} \cdot \frac{1}{\cos 25^\circ} = \frac{-\cos (90^\circ - 40^\circ)}{\cos 40^\circ} \cdot \frac{1}{\cos 25^\circ} = \frac{-\sin 40^\circ}{\cos 40^\circ} \cdot \frac{1}{\cos 25^\circ} \][/tex]
Using a known trigonometric identity, the final simplified form is:
[tex]\[ \boxed{-\tan 40^\circ \cdot \sec 25^\circ} \][/tex]
---
Instruction IV: Work out the following question showing all the necessary steps.
24. If the measures of all four interior angles of a cyclic quadrilateral add up to 360 degrees and are given as [tex]\((3x-15)^{\circ}, (2x+10)^{\circ}, (4x-8)^{\circ},\)[/tex] and [tex]\((6x-17)^{\circ}\)[/tex], solve for [tex]\(x\)[/tex]:
- Sum the angles: [tex]\((3x - 15) + (2x + 10) + (4x - 8) + (6x - 17) = 360\)[/tex]
- Simplify by combining like terms and solving for [tex]\(x\)[/tex]:
[tex]\[ 3x + 2x + 4x + 6x - 15 + 10 - 8 - 17 = 360 \][/tex]
[tex]\[ 15x - 30 = 360 \][/tex]
[tex]\[ 15x = 390 \][/tex]
[tex]\[ x = 26 \][/tex]
25. Solve the quadratic equation [tex]\(2x^2 - 6x + 8 = 0\)[/tex]:
Let's rewrite and solve it:
[tex]\[ 2x^2 - 6x + 8 = 0 \][/tex]
Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 2\)[/tex], [tex]\(b = -6\)[/tex], and [tex]\(c = 8\)[/tex]:
[tex]\[ b^2 - 4ac = (-6)^2 - 4(2)(8) = 36 - 64 = -28 \][/tex]
Since the discriminant ([tex]\(\Delta\)[/tex]) is [tex]\(\boxed{-28}\)[/tex] the equation has complex solutions.
26. A ladder leaning against a wall forms an angle of [tex]\(30^{\circ}\)[/tex] with the wall. If the foot of the ladder is 8 m from the wall, find the length of the ladder [tex]\(L\)[/tex]:
Using trigonometry, specifically the sine function:
[tex]\(\sin(30^\circ) = \frac{8}{L}\)[/tex], where [tex]\(\sin 30^\circ = 0.5\)[/tex]:
[tex]\[ 0.5 = \frac{8}{L} \][/tex]
[tex]\[ L = 16 \, \text{meters} \][/tex]
27. Two chords [tex]\(AB\)[/tex] and [tex]\(CD\)[/tex] of a circle intersect at right angles at a point inside the circle. If [tex]\(\angle BAC = 35^\circ\)[/tex], find [tex]\(\angle ABD\)[/tex]:
Since the angles formed by intersecting chords (perpendicular) sum up to [tex]\(90^\circ\)[/tex]:
[tex]\[ \angle BAC + \angle ABD = 90^\circ \][/tex]
[tex]\[ 35^\circ + \angle ABD = 90^\circ \][/tex]
[tex]\[ \angle ABD = 55^\circ \][/tex]
28. Solve the equation [tex]\(2x^2 + x - 1 = 0\)[/tex] in the interval [tex]\([0, 2]\)[/tex]:
Given the discriminant and solutions:
[tex]\[ D = 9, x_1 = 0.5, x_2 = -1.0, \quad \text{solutions in } [0, 2] = [0.5] \][/tex]
Therefore, the solution set for the equation in the interval [tex]\([0, 2]\)[/tex] is:
[tex]\[ \boxed{[0.5]} \][/tex]
This completes the detailed steps for each given question.
