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2 years ago

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The admission fee at a movie theater is $5 for children and $9 for adults. If 3200 people go to the movies and $24000 is collect

ed, how many adults went to the movies?

1 answer:

Usimov [2.4K]2 years ago

6 0

For this problem, let x be the number of children and y for adults. Formulate the equations: 1st equation, x + y = 3,200 and 2nd equation 5x + 9y = 24,000. Re-arrange 1st equation into x = 3200 - y. Then, substitute into 2nd equation, 5(3,200-y) + 9y = 24,000. Then, solve for y. The 16,000 - 5y + 9y = 24000. Final answer is, y = 2000 adults went to watch the movie.

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Question:

A Sunday newspaper lists the 10 used car prices for a foreign compact, with age x measured in years and selling price y measured in thousands of dollars. A Sunday newspaper lists the 10 used car prices for a foreign compact, with age x measured in years and selling price y measured in thousands of dollars.

Picture of data can be found in the picture attached.

Answer:

Slope = -1.71

Intercept = 22.05

Step-by-step explanation:

Equation of a line usually takes the general form :

ŷ = mx + c

Where :

m = slope or gradient

c = intercept

Inputting the data into a linear regression calculator ;

The output is : ŷ = -1.70872X + 22.04985

Comparing the output to the general formula;

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In triangle XYZ, if angle X is congruent to angle Z, XY = 13x - 21, YZ = 8x - 6, & XZ = x + 4, find x & the measure of e

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Answer:

Given: In triangle XYZ, $\angle X \cong \angle Z$ , XY=13x-21 , YZ=8x-6 and XZ=x+4.

If two angles are congruent if and only if, they measure the same number of degrees.

therefore, from the given condition; $\angle X = \angle Z$ .

Isosceles Triangle : An triangle with two equal sides. The angles opposite the equal sides are also equal.

Therefore, the triangle XYZ is an isosceles triangle, as the base angle $\angle X$ and $\angle Z$ are equal and the sides opposite to these angles are also equal i.e, XY=YZ.

Since, we have XY=YZ ,

⇒ 13x-21 =8x-6 or

13x-8x=21-6 or

5x=15

Simplify:

x=3

Therefore, the sides of an isosceles triangle are:

XY = 13x-21 = $13\cdot 3 -21$ = 39-21 = 18 unit ,

YZ = 8x-6 = $8 \cdot3 -4$ = 24-6 =18 unit,

and XZ = x+4 = 3+4 =7 unit.

Since, the triangle is isosceles, we draw a line from an vertex angle Y of a triangle XYZ which is perpendicular (i.e, of 90 degree angle) to opposite side meets the opposite side at its midpoint i.e, say W.

As W is the midpoint(i.e, it divide the sides into two equal halves),

XW=WZ = $\frac{7}{2} =3.5$ unit

Now, use the Pythagorean theorem, to find the altitude WY.

⇒ $(WY)^2+(WZ)^2=(YZ)^2$

Substitute the value of WZ = 3.5 unit and YZ = 18 unit in above formula to calculate the length of WY;

⇒ $(WY)^2+(3.5)^2=(18)^2$

Simplify:

$(WY)^2=324-12.25=311.75$ or

$WY=\sqrt{311.75}$

On simplify we get;

WY=17.65 unit(approx.)

The vertex angle Y is split into two equal angles, we can find the vertex angle by finding the one of the base angle by using the fact; $Cosine = \frac{Base}{Hypotenuse}$ .

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⇒ $\cos Z = 0.195$ (approx) or

$Z=\cos^{-1}(0.195)$

Therefore, the angle $\angle Z = 78.7^{\circ}$ .

The sum of the measure of the angle in the triangle is 180 degree.

In triangle XYZ.

$\angle X +\angle Y+\angle Z =180^{\circ}$

Since $\angle X=\angle Z =78.7^{\circ}$

then,

$78.7^{\circ}+\angle Y+78.7^{\circ}=180^{\circ}$

⇒ $157.4^{\circ}+\angle Y=180^{\circ}$

Simplify:

$\angle Y=22.6^{\circ}$ .

Therefore, the measure of each angle are: $\angle X=\angle Z=78.7^{\circ}$ and $\angle Y =22.6^{\circ}$

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For this case we have the following system of equations:

$x-3y = -13\\5x + 7y = 34$

From the first equation we clear "x":

$x = -13 + 3y$

We substitute in the second equation:

$5 (-13 + 3y) + 7y = 34$

We apply distributive property:

$-65 + 15y + 7y = 34$

We add similar terms:

$-65 + 22y = 34$

We add 65 to both sides:

$22y = 34 + 65\\22y = 99$

We divide between 22 on both sides:

$y = \frac {99} {22}\\y = \frac {9} {2}$

We look for the value of the variable "x":

$x = -13 + 3 \frac {9} {2}\\x = -13 + \frac {27} {2}\\x = \frac {-26 + 27} {2}\\x = \frac {1} {2}$

Thus, the solution of the system is:

$(x, y): (\frac {1} {2}, \frac {9} {2})$

ANswer:

$(x, y): (\frac {1} {2}, \frac {9} {2})$

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