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2 years ago

If PQR = STU, find the measure of each part.

Mathematics

1 answer:

alina1380 [7]2 years ago

6 0

The answer is below

Step-by-step explanation:

Two triangles are said to be congruent if for the two triangles all the three sides are equal and all the three angles are equal to each other.

Given that ΔPQR is congruent to ΔSTU, hence all corresponding sides and corresponding angles are equal to each other.

∠P = ∠S = 59°

∠S + ∠T + ∠U = 180° (sum of angles in a triangle)

59 + 75 + ∠U = 180°

134 + ∠U = 180°

∠U = 46°

∠U = ∠R = 46°

∠Q = ∠T = 75°

Also, for the sides:

ST = PQ = 7 m

SU = PR = 8 m

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x = 0.53 cm

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Step-by-step explanation:

Refer to the attached diagram:

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The length of the shaded region is given by

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So, the volume of the box becomes,

$V = \frac{1}{2} (5 - 3x) \times (3 - 2x) \times x \\\\V = \frac{1}{2} (5 - 3x) \times (3x - 2x^2) \\\\V = \frac{1}{2} (15x -10x^2 -9 x^2 + 6 x^3) \\\\V = \frac{1}{2} (6x^3 -19x^2 + 15x) \\\\$

In order to maximize the volume enclosed by the box, take the derivative of volume and set it to zero.

$\frac{dV}{dx} = 0 \\\\\frac{dV}{dx} = \frac{d}{dx} ( \frac{1}{2} (6x^3 -19x^2 + 15x)) \\\\\frac{dV}{dx} = \frac{1}{2} (18x^2 -38x + 15) \\\\\frac{dV}{dx} = \frac{1}{2} (18x^2 -38x + 15) \\\\0 = \frac{1}{2} (18x^2 -38x + 15) \\\\18x^2 -38x + 15 = 0 \\\\$

We are left with a quadratic equation.

We may solve the quadratic equation using quadratic formula.

The quadratic formula is given by

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Where

$a = 18 \\\\b = -38 \\\\c = 15 \\\\$

$x=\frac{-(-38)\pm\sqrt{(-38)^2-4(18)(15)}}{2(18)} \\\\x=\frac{38\pm\sqrt{(1444- 1080}}{36} \\\\x=\frac{38\pm\sqrt{(364}}{36} \\\\x=\frac{38\pm 19.078}{36} \\\\x=\frac{38 + 19.078}{36} \: or \: x=\frac{38 - 19.078}{36}\\\\x= 1.59 \: or \: x = 0.53 \\\\$