25.02.2023

A triangle has sides with lengths of 63 miles, 64 miles, and 85 miles. Is it a right triangle? Yes or no?

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09.07.2023, solved by verified expert
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no it is not a right angled triangle.

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forthetra


A triangle has sides with lengths of 63 miles,, №18011122, 25.02.2023 14:16
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Mathematics
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no it is not a right angled triangle.

Step-by-step explanation:

forthetra


A triangle has sides with lengths of 63 miles, 64 miles, and 85 miles. Is it a right triangle? Yes o
Mathematics
Step-by-step answer
P Answered by PhD

1. It is shifted 2 units down.

The graph of y=-8x^2 -2 is shifted 2 units down with respect to the graph of y=-8x^2. We can prove this by taking, for instance, x=0, and calculating the value of y in the two cases. In the first function:

y=-8*0^2 -2=-2

In the second function:

y=-8*0^2 =0

So, the first graph is shifted 2 units down.

2. 160.56 m

The path of the rocket is given by:

y=-0.06 x^2 +9.6 x +5.4

The problem asks us to find how far horizontally the rocket lands - this corresponds to find the value of x at which the height is zero: y=0. This means we have to solve the following equation

-0.06 x^2 +9.6 x+5.4 =0

Using the formula,

x=\frac{-9.6 \pm \sqrt{(9.6)^2-4(-0.06)(5.4)}}{2(-0.06)}

which has two solutions: x_1 = 160.56 m and x_2 = -0.56 m. The second solution is negative, so it has no physical meaning, therefore the correct answer is 160.56 m.

3. 27.43 m

The path of the rock is given by:

y=-0.02 x^2 +0.8 x +37

The problem asks us to find how far horizontally the rock lands - this corresponds to find the value of x at which the height is zero: y=0. This means we have to solve the following equation

-0.02 x^2 +0.8 x+37 =0

Using the formula,

x=\frac{-0.8 \pm \sqrt{(0.8)^2-4(-0.02)(37)}}{2(-0.02)}

which has two solutions: x_1 = 67.43 m and x_2 = -27.43 m. In this case, we have to choose the second solution (27.43 m), since the rock was thrown backward from the initial height of 37 m, so the negative solution corresponds to the backward direction.

4. (-2, 16) and (1, -2)

The system is:

y=x^2 -5x +2 (1)

y=-6x+4 (2)

We can equalize the two equations:

x^2 -5x+2 = -6x +4

which becomes:

x^2 + x -2 =0

Solving it with the formula, we find two solutions: x=-2 and x=1. Substituting both into eq.(2):

x=-2 --> y=-6 (-2) +4 = 12+4 = 16

x=1 --> y=-6 (1) +4 = -6+4 =-2

So, the solutions are (-2, 16) and (1, -2).

5. (-1, 1) and (7, 33)

The system is:

y=x^2 -2x -2 (1)

y=4x+5 (2)

We can equalize the two equations:

x^2 -2x-2 = 4x +5

which becomes:

x^2 -6x -7 =0

Solving it with the formula, we find two solutions: x=7 and x=-1. Substituting both into eq.(2):

x=7 --> y=4 (7) +5 = 28+5 = 33

x=-1 --> y=4 (-1) +5 = -4+5 =1

So, the solutions are (-1, 1) and (7, 33).

6. 2.30 seconds

The height of the object is given by:

h(t)=-16 t^2 +85

The time at which the object hits the ground is the time t at which the height becomes zero: h(t)=0, therefore

-16t^2 +85 =0

By solving it,

16t^2 = 85

t^2 = \frac{85}{16}

t=\sqrt{\frac{85}{16}}=2.30 s

7. Reaches a maximum height of 19.25 feet after 0.88 seconds.

The height of the ball is given by

h(t)=-16t^2 + 28t + 7

The vertical velocity of the ball is equal to the derivative of the height:

v(t)=h'(t)=-32t+28

The maximum height is reached when the vertical velocity becomes zero: v=0, therefore when

-32t + 28 =0

from which we find t=0.88 s

And by substituting these value into h(t), we find the maximum height:

h(t)=-16(0.88)^2 + 28(0.88) + 7 = 19.25 m

8. Reaches a maximum height of 372.25 feet after 4.63 seconds.

The height of the boulder is given by

h(t)=-16t^2 + 148t + 30

The vertical velocity of the boulder is equal to the derivative of the height:

v(t)=h'(t)=-32t+148

The maximum height is reached when the vertical velocity becomes zero: v=0, therefore when

-32t + 148 =0

from which we find t=4.63 s

And by substituting these value into h(t), we find the maximum height:

h(t)=-16(4.63)^2 + 148(4.63) + 30 = 372.25 m

9. 12 m

Let's call x the length of the side of the original garden. The side of the new garden has length (x+3), so its area is

(x+3)^2 = 225

Solvign this equation, we find

x+3 = \sqrt{225}=15

x=15-3=12 m

10. 225/4

In fact, if we write x^2 +15 x + \frac{225}{4}, we see this is equivalent to the perfect square:

(x+\frac{15}{2})^2 = x^2 +15 x +\frac{225}{4}

11. -11.56, 1.56

The equation is:

x^2 +10 x -18 =0

By using the formula:

x=\frac{-10 \pm \sqrt{(10)^2-4(1)(-18)}}{2*1}

which has two solutions: x=-11.56 and 1.56.

12. -10.35, 1.35

The equation is:

x^2 +9 x -14 =0

By using the formula:

x=\frac{-9 \pm \sqrt{(9)^2-4(1)(-14)}}{2*1}

which has two solutions: x=-10.35 and 1.35.

Mathematics
Step-by-step answer
P Answered by PhD

1. It is shifted 2 units down.

The graph of y=-8x^2 -2 is shifted 2 units down with respect to the graph of y=-8x^2. We can prove this by taking, for instance, x=0, and calculating the value of y in the two cases. In the first function:

y=-8*0^2 -2=-2

In the second function:

y=-8*0^2 =0

So, the first graph is shifted 2 units down.

2. 160.56 m

The path of the rocket is given by:

y=-0.06 x^2 +9.6 x +5.4

The problem asks us to find how far horizontally the rocket lands - this corresponds to find the value of x at which the height is zero: y=0. This means we have to solve the following equation

-0.06 x^2 +9.6 x+5.4 =0

Using the formula,

x=\frac{-9.6 \pm \sqrt{(9.6)^2-4(-0.06)(5.4)}}{2(-0.06)}

which has two solutions: x_1 = 160.56 m and x_2 = -0.56 m. The second solution is negative, so it has no physical meaning, therefore the correct answer is 160.56 m.

3. 27.43 m

The path of the rock is given by:

y=-0.02 x^2 +0.8 x +37

The problem asks us to find how far horizontally the rock lands - this corresponds to find the value of x at which the height is zero: y=0. This means we have to solve the following equation

-0.02 x^2 +0.8 x+37 =0

Using the formula,

x=\frac{-0.8 \pm \sqrt{(0.8)^2-4(-0.02)(37)}}{2(-0.02)}

which has two solutions: x_1 = 67.43 m and x_2 = -27.43 m. In this case, we have to choose the second solution (27.43 m), since the rock was thrown backward from the initial height of 37 m, so the negative solution corresponds to the backward direction.

4. (-2, 16) and (1, -2)

The system is:

y=x^2 -5x +2 (1)

y=-6x+4 (2)

We can equalize the two equations:

x^2 -5x+2 = -6x +4

which becomes:

x^2 + x -2 =0

Solving it with the formula, we find two solutions: x=-2 and x=1. Substituting both into eq.(2):

x=-2 --> y=-6 (-2) +4 = 12+4 = 16

x=1 --> y=-6 (1) +4 = -6+4 =-2

So, the solutions are (-2, 16) and (1, -2).

5. (-1, 1) and (7, 33)

The system is:

y=x^2 -2x -2 (1)

y=4x+5 (2)

We can equalize the two equations:

x^2 -2x-2 = 4x +5

which becomes:

x^2 -6x -7 =0

Solving it with the formula, we find two solutions: x=7 and x=-1. Substituting both into eq.(2):

x=7 --> y=4 (7) +5 = 28+5 = 33

x=-1 --> y=4 (-1) +5 = -4+5 =1

So, the solutions are (-1, 1) and (7, 33).

6. 2.30 seconds

The height of the object is given by:

h(t)=-16 t^2 +85

The time at which the object hits the ground is the time t at which the height becomes zero: h(t)=0, therefore

-16t^2 +85 =0

By solving it,

16t^2 = 85

t^2 = \frac{85}{16}

t=\sqrt{\frac{85}{16}}=2.30 s

7. Reaches a maximum height of 19.25 feet after 0.88 seconds.

The height of the ball is given by

h(t)=-16t^2 + 28t + 7

The vertical velocity of the ball is equal to the derivative of the height:

v(t)=h'(t)=-32t+28

The maximum height is reached when the vertical velocity becomes zero: v=0, therefore when

-32t + 28 =0

from which we find t=0.88 s

And by substituting these value into h(t), we find the maximum height:

h(t)=-16(0.88)^2 + 28(0.88) + 7 = 19.25 m

8. Reaches a maximum height of 372.25 feet after 4.63 seconds.

The height of the boulder is given by

h(t)=-16t^2 + 148t + 30

The vertical velocity of the boulder is equal to the derivative of the height:

v(t)=h'(t)=-32t+148

The maximum height is reached when the vertical velocity becomes zero: v=0, therefore when

-32t + 148 =0

from which we find t=4.63 s

And by substituting these value into h(t), we find the maximum height:

h(t)=-16(4.63)^2 + 148(4.63) + 30 = 372.25 m

9. 12 m

Let's call x the length of the side of the original garden. The side of the new garden has length (x+3), so its area is

(x+3)^2 = 225

Solvign this equation, we find

x+3 = \sqrt{225}=15

x=15-3=12 m

10. 225/4

In fact, if we write x^2 +15 x + \frac{225}{4}, we see this is equivalent to the perfect square:

(x+\frac{15}{2})^2 = x^2 +15 x +\frac{225}{4}

11. -11.56, 1.56

The equation is:

x^2 +10 x -18 =0

By using the formula:

x=\frac{-10 \pm \sqrt{(10)^2-4(1)(-18)}}{2*1}

which has two solutions: x=-11.56 and 1.56.

12. -10.35, 1.35

The equation is:

x^2 +9 x -14 =0

By using the formula:

x=\frac{-9 \pm \sqrt{(9)^2-4(1)(-14)}}{2*1}

which has two solutions: x=-10.35 and 1.35.

Mathematics
Step-by-step answer
P Answered by PhD

Cost of 7 gallons=$24.50

Cost of 1 gallon=24.50/7=3.5

Cost of 15 gallons=15*3.5=52.5

Cost of 15 gallons will be $52.5

Mathematics
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P Answered by PhD

For 1 flavor there are 9 topping

Therefore, for 5 different flavors there will be 5*9 choices

No of choices= 5*9

=45 

Mathematics
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P Answered by PhD

The answer is in the image 

The answer is in the image 
Mathematics
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P Answered by PhD

The wood before starting =12 feet

Left wood=6 feet

Wood used till now=12-6=6 feet

Picture frame built till now= 6/(3/4)

=8 pieces

Therefore, till now 8 pieces have been made.

Mathematics
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P Answered by PhD

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