Why is estimate usually greater than or less than the exact sum of two numbers?

What is the slope of the line shown in the graph?

iVinArrow [24]

Answer:

t

Step-by-step explanation:

6 0

2 years ago

Plssssssssssssssssssss Answer this is Major?

julsineya [31]

Answer:

See below because there are 9 parts (A through I)

Explanation:

Part A: write an equation that models the area of the figure. Let y represent the area, and write your answer in the form y = ax2 + bx + c.

The figure shows a rectangular table with these dimensions:

Length: - x + 64
Witdth: x + 4

The area of a rectangle is width × length:

$(x + 4)\times (-x+64)$

Use distributive property:

$x\cdot (-x)+x\cdot(64)+4\cdot (-x)+4\cdot (64)=-x^2+64x-4x+256$

Simplify:

$-x^2+64x-4x+256=-x^2+60x+256$

Part B. Graph the equation you wrote in part A. Adjust the zoom of the graphing window so the vertex, x-intercepts, and y-intercept can be seen.

1. Factor the equation:

Common factor - 1:

$-x^2+60x+256=-(x^2-60x-256)$

Find two numbers that add - 60 and whose product is -256. Theyb are -64 and + 4

$-(x-64)(x+4)$

2. Find the roots:

Equal the expression to zero:

$-(x-64)(x+4)=0\\ \\ x-64=0\implies x=64\\ \\ x+4=0\implies x=-4$

Those are the x-intercepts: (-4,0) and (64,0)

3. Find the symmetry axis:

The simmetry axis is the line x = the middle value between the two roots:

$x=(64-4)/2=60/2=30$

4. Find the vertex

The vertex has x-coordinate equal to the x axis (30 in this case).

Substitute in the equation of find the y-coordinate:

$y=-(30-64)(30+4)=-(-34)(34)=1,156$

Hence, the vertex is (30, 1,156)

5. Find the y-intercept

Make x = 0

$y=-(x^2-60x-256)=-(0-256)=256$

Hence, the y-intercept is (0, 256)

With the x-incercepts, the y-intercept, the axis of symmetry, and the vertex, you can sketch the graph.

You can see now the graph in the attached figure

Part C. Extreme location of the graph

The graph shows that the parabola opens downward. That is due to the fact that the coefficient of the leading term (x²) is negative.

The parabola starts in the second quadrant. starts growing, crosses the x-axis at (-4,0), crosses the y-axis at (0,256), reaches the maximum value at (30, 1156), and then decreases toward the fouth quadrant, crossing the x-axis at (64,0).

Thus the vertex is a maximun, and the coordinates of the maximum are (30, 1156).

Part D. According to the graph, what is the maximum possible area of the game board? Give your answer to the nearest whole number. (Assume that the maximum area is not reduced by the open hole in the game board.)

The maximum possible area of the game is the maximum value of the function y = -x² + 60x + 256.

This value was calculated as y = 1156.

Part E. Use the original expressions for the length and width, and substitute the x-coordinate from the extreme location. What are the length and width of the game board at the extreme location?

The length is:

length = - x + 64 inches

x = 30

length = - 30 + 64 = 34 inches

The width is:

width = x + 4

x = 30

width = 30 + 4 = 34 inches

Part F. What type of quadrilateral will be formed when the game board covers the maximum possible area?

Since the length and the width are equal, the quadrilateral is a square.

Part G. Suppose the carnival director asks you to create a game board that is 1,120 square inches. Find the dimensions that would meet this request by setting the area equation equal to 1,120, solving for x, and substituting x into the expressions for the length and width.

$y=-x^2+60x+256\\ \\ 1,120=-x^2+60x+256\\ \\ x^2-60x-256+1120=0\\ \\ x^2-60x+864=0$

Factor:

Find two numbers whose sum is - 60 and the product os 864. They are -24 and - 34:

$x^2-60x+864=(x-24)(x-36)$

Use the zero product rule:

$(x-24)(x-36)=0\\ \\ x-24=0\implies x=24\\ \\ x-36=0\implies x=36$

Now substitute to find the dimensions:

x = 36

length = - x + 64

length = - 36 + 64 = 28

width = x + 4 = 36 + 4 = 40

Hence, legth = 28, width = 40

x = 24

length = - x + 64 = -24 + 64 = 40

width = x + 4 = 24 + 4 = 28

Part H. When you solved the area equation for x, did any extraneous solutions result? Describe how an extraneous solution would arise in this situation.

The two solutions are valid (non extraneous) because both leads to positive real dimensions for which the areas can be 1,120 in².

28×40 = 1,120

40×28 = 1,120

An extraneous solution could arise if you try to find areas for which x is greater than or equal to 64, because in that case - x + 64 would be zero or negative and dimensions must be positive.

For the same reason, also an extraneous solution would arise if you try to fix areas for which x is less than or equal to - 4.

So, the domain of your function has to be - 4 < x < 64.

Part I. What method of solving quadratics did you use to solve the equation set equal to 1,120? Why did you choose this method?

The method use was factoring.

Discuss the usefulness of other methods of solving quadratics as they pertain to this scenario.

The other importants methods are graphical and the quadratic equation.

For graphical method you graph your parabola and find the values of x that sitisfies the area searched (value of y).

The quadratic equation gives the y-values (areas) without factoring:

$\frac{-b+/-\sqrt{b^2-4(a)(c)} }{2(a)}$

8 0

3 years ago

HELP ME ASAP PLEASE!!!!

vesna_86 [32]

Answer:

D good luck

Step-by-step explanation:

6 0

3 years ago

The set (3,5,___________ could not be the sides of a triangle 3,2,5,7

inna [77]

Answer:

(3, 5, 2)

Step-by-step explanation:

Many authors interpret the triangle inequality to mean the sum of the two short sides must exceed the length of the long side. For side measures 2, 3, 5, the sum of the two short sides is exactly equal to the long side, in violation of the triangle inequality. Hence (3, 5, 2) is not a triangle.

___

Comment on the triangle inequality

Other authors allow the "or equal to" case, meaning sides of lengths 2, 3, 5 will be considered to be a triangle because 2+3=5. This interpretation of the triangle inequality will result in no solution to your question.

A (3, 5, 2) "triangle" will look like a line segment of length 5. It will have an area of zero.

3 0

4 years ago

ASAP!! Please help me. I will not accept nonsense answers, but will mark as BRAINLIEST if you answer is correctly with solutions

ryzh [129]

Answer:

The third option is correct.

The domain of the function is given by

$Domain = 0 \leq t \leq 40$

The range of the function is given by

$Range = 0 \leq V(t) \leq 200$

Step-by-step explanation:

Sara bought a cell phone for $200 and its value has decreased at rate of $5 per month.

V(t) is the value of the phone and t is the number of months.

The domain of the function is the possible values of the number of months t.

The domain of the function is given by

$Domain = 0 \leq t \leq 40$

The range of the function is the values of V(t) that we get after substituting the possible values of the number of months t.

The range of the function is given by

$Range = 0 \leq V(t) \leq 200$

When the number of months is t = 0 then the value of the function is maximum V(t) = 200

When the number of months is t = 40 then the value of the function is minimum V(t) = 0

Therefore, the third option is correct.

6 0

4 years ago